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John–Löwner ellipsoids and entropy of multiplier operators on rank A. K. Kushpel Department of Mathematics, Çankaya University, Ankara, Turkey
Bibliographic databases:
    § 1. IntroductionThe notion of entropy was introduced by Kolmogorov [14]. Let $X$ and $Y$ be Banach spaces with unit balls $B_{X}$ and $B_{Y}$, respectively. For a compact set $A\subset Y$ we define the entropy number $e_{n}(A,Y)$ as the infimum of all $\varepsilon >0$ such that there exist $\{y_{k}\}_{k=1}^{2^{n-1}}\subset Y$ such that
$$
\begin{equation*}
A\subset \bigcup_{k=1}^{2^{n-1}}(y_{k}+\varepsilon B_{Y}),
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
e_{n}(A,Y)=\inf_{\varepsilon >0}\biggl\{ A\subset \bigcup_{k=1}^{2^{n-1}}(y_{k}+\varepsilon B_{Y}) \text{ for some } \{ y_{k}\}_{k=1}^{2^{n-1}}\subset Y\biggr\}.
\end{equation*}
\notag
$$
Let $v\colon X\to Y$ be a compact operator. Then the $n$th entropy number $e_{n}(v):=e_{n}(v\colon X\to Y)$ is the infimum over all positive $\varepsilon$ such that there exist $\{y_{k}\}_{k=1}^{2^{n-1}}\subset Y$ such that
$$
\begin{equation*}
v(B_{X}) \subset \bigcup_{k=1}^{2^{n-1}}(y_{k}+\varepsilon B_{Y}).
\end{equation*}
\notag
$$
The notion of entropy can be given in terms of cowidths. Let $(X,\|\cdot\|)$ be a Banach space with unit ball $B_{X}$, let $A\subset X$, let $Z$ be a coding set and $\Phi$ be a family of mappings $\phi \colon A\to Z$. Let be the preimage of $x\in Z$, and let
$$
\begin{equation*}
\operatorname{diam}(A)=\sup_{x,y\in A}\|x-y\|, \qquad A\subset X.
\end{equation*}
\notag
$$
The corresponding cowidths can be defined by
$$
\begin{equation*}
\mathrm{co}^{\Phi}(A,X)=\inf_{\phi \in \Phi}\sup_{x\in A}\operatorname{diam}\{ \phi^{-1}(\phi (x)) \}.
\end{equation*}
\notag
$$
In particular, if $Z$ is the set of all $N$-point sets $\{x_{k}\}_{k=1}^{N}\subset X$ and $\Phi$ is the set of all maps $\phi\colon A\to \{x_{k}\}_{k=1}^{N}$, then $\mathrm{co}^{\Phi}(A,X)$ is called the entropy width $\varepsilon_{N}(A,X)$. If $N=2^{n-1}$, $n\in \mathbb{N}$, then
$$
\begin{equation*}
e_{n}(A,X) \leqslant \varepsilon_{N}(A,X) \leqslant 2e_{n}(A,X).
\end{equation*}
\notag
$$
Suppose that $A$ is a compact origin symmetric subset of a Banach space $X$. Then the linear $n$-width of $A$ in $X$ is defined by
$$
\begin{equation*}
\delta_{n}(A,X)=\inf_{P_{n}}\sup_{f\in A}\|f-P_{n}f\|,
\end{equation*}
\notag
$$
where $P_{n}$ ranges over all linear operators of rank at most $n$ that map $X$ into itself. Let $Y$ be a Banach space, $u\colon Y \to X$ be a compact linear operator, and let $A=uB_{Y}$. Assume that $f(m)\colon \mathbb{N}\to \mathbb{R}$ is a positive function increasing (or nondecreasing) monotonically for large $m\in \mathbb{N}$ and such that for some fixed $C_{0}$ and each $j\in\mathbb{N}$. Then there exists a positive constant $C$ such that for all $m\in \mathbb{N}$ we have
$$
\begin{equation}
\sup_{1\leqslant n\leqslant m}f(n) e_{n}(A,X) \leqslant C\sup_{1\leqslant n\leqslant m}f(n) \delta_{n-1}(A,X)
\end{equation}
\tag{1.2}
$$
(see [34], for example). In the present paper we investigate the entropy of sets of smooth functions induced by multiplier operators on compact globally symmetric connected spaces of rank $1$ (or two-point homogeneous spaces). A complete classification of the two-point homogeneous spaces $\mathbb{M}^{d}$, $\dim \mathbb{M}^{d}=d >1$, was presented in [35]. They are the real spheres $\mathbb{S}^{d}$, $d=2,3, \dots$, the real projective spaces $\mathrm{P}^{d}(\mathbb{R})$, $d=2,3, \dots$, the complex projective spaces $\mathrm{P}^{d}(\mathbb{C})$, $d=4,6, \dots$, the quaternion projective spaces $\mathrm{P}^{d}(\mathbb{H})$, $d=8,12,\dots$, and the Cayley elliptic plane $\mathrm{P}^{16}(\mathrm{Cay})$. For more information on this classification, see [5], [7], [11] and [12], for example. In § 2 we consider some properties of two-point homogeneous spaces which are important for our applications. In particular, we present formulae in closed form for the dimensions of direct sums of eigenspaces of Laplace–Beltrami operators. Our analysis is based on a detailed study of convex origin-symmetric bodies $V$ in $\mathbb{R}^{n}$ induced by eigenfunctions of the Laplace–Beltrami operators on homogeneous manifolds. Namely, we establish estimates for the volumes of the John–Löwner ellipsoids $\mathcal{E}(V)$ (ellipsoids of maximum volume contained in $V$) to find order-sharp estimates for entropy numbers, since entropy is a natural analogue of volume in infinite-dimensional Banach spaces (see [31] and [33], for example). Observe that in the case of the trigonometric system (when $\phi_{k}(t)=(2\pi)^{-d/2}\exp (i\langle k,t\rangle)$, $k\in \mathbb{Z}^{d}$, $t\in\mathbb{T}^{d}$) the corresponding estimates were obtained in [13]. However, the possibilities of the method used there are essentially limited to the case of the trigonometric system on the torus. It is underlined in [33], Ch. 7, that even in the case of the trigonometric system the problem of estimating the corresponding volumes is highly nontrivial. In the case of two-point homogeneous spaces volume estimates are connected with a range of difficulties of fundamental nature in comparison with the torus. In particular, representations of shifts of trigonometric functions on the torus $\mathbb{T}^{d}$ (used in Lemma 2.1 in [13] to estimate the volumes of the corresponding John–Löwner ellipsoids) are very simple:
$$
\begin{equation*}
\exp (i\langle k,t+h\rangle)=\exp (i\langle k,h\rangle) \exp (i\langle k,t\rangle) \quad \forall \,t,h\in \mathbb{T}^{d}, \quad k\in \mathbb{Z}^{d}.
\end{equation*}
\notag
$$
The situation changes completely in the case of two-point homogeneous spaces. In particular, even in the case of $\mathbb{S}^{2}$, after applying a rotation to a spherical harmonic of degree $k$, this rotated spherical harmonic can be expressed as a linear combination of the spherical harmonics $Y_{j}^{k}$ of degree $k$, with coefficients given by a fairly complicated Wigner $D$-matrix. The approach we present allows us to overcome this range of difficulties. Namely, in § 3, we find the entropy of the Sobolev classes $W_{p}^{\gamma }(\mathbb{M}^{d})$ in $L_{q}(\mathbb{M}^{d})$, that is, we show that
$$
\begin{equation*}
e_{n}(W_{p}^{\gamma}(\mathbb{M}^{d}),L_{q}(\mathbb{M}^{d})) \asymp n^{-{\gamma}/{d}}, \qquad \gamma >0, \quad 1\leqslant q\leqslant p\leqslant \infty.
\end{equation*}
\notag
$$
These results improve the estimates obtained in the case of arbitrary compact homogeneous Riemannian manifolds (see Remark 3.7 in [25]), namely,
$$
\begin{equation*}
e_{n}(W_{\infty}^{\gamma}(\mathbb{M}^{d}),L_{1}(\mathbb{M}^{d})) \gg n^{-\gamma/d}\log^{-1}n, \qquad \gamma >0,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
e_{n}(W_{\infty}^{\gamma}(\mathbb{M}^{d}),L_{1}(\mathbb{M}^{d})) \ll n^{-\gamma/d}, \qquad \gamma >0.
\end{equation*}
\notag
$$
The last line follows from (1.2), (2.37) and (2.39). More precisely, put $f(n)=n^{\gamma/d}$ in (1.2). Then condition (1.1) is satisfied, and using an embedding we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sup_{1\leqslant n\leqslant m}n^{\gamma/d}e_{n}(W_{\infty}^{\gamma}(\mathbb{M}^{d}),L_{1}(\mathbb{M}^{d})) \leqslant C\sup_{1\leqslant n\leqslant m}n^{\gamma/d}\delta_{n-1}(W_{\infty}^{\gamma}(\mathbb{M}^{d}),L_{1}(\mathbb{M}^{d})) \\ &\qquad \leqslant C\sup_{1\leqslant n\leqslant m}n^{\gamma/d}\delta_{n-1}(W_{2}^{\gamma}(\mathbb{M}^{d}),L_{2}(\mathbb{M}^{d})) \leqslant C_{1}. \end{aligned}
\end{equation*}
\notag
$$
Note that in [25] we used a different approach, developed in the cycle of works [15]–[22] and [24]. Observe that the entropy numbers of Sobolev classes on the $N$-dimensional cube were considered in [1]. In the case of $\mathbb{T}^{d}$ order-sharp lower bounds for the entropy numbers in $L_{1}$ of the Sobolev classes $W^{\gamma}_{\infty}$ were obtained in [13]. Various volume estimates of origin-symmetric convex bodies associated with orthonormal systems on manifolds were established in [27]. In this article positive constants are mostly denoted by $C$. We do not carefully distinguish between different constants, neither do we try to find good estimates for them. The same character is used to denote different constants in different parts of the article. For easy notation we write $a_{n}\gg b_{n}$ for two sequences if $a_{n}\geqslant Cb_{n}$ for some $C>0$ and each $n\in \mathbb{N}$, and $a_{n}\asymp b_{n}$ if $C_{1}b_{n}\leqslant a_{n}\leqslant C_{2}b_{n}$ for all $n\in \mathbb{N}$ and some positive constants $C_{1}$ and $C_{2}$. In some cases the constant $C(d)$ can depend on the dimension $d$, which is always assumed to be fixed. § 2. Harmonic analysisLet $\mathbb{M}^{d}$ be a two-point homogeneous Riemannian manifold. Each such manifold $\mathbb{M}^{d}$ can be represented as the orbit space of a certain subgroup $\mathcal{H}$ of the orthogonal group $\mathcal{G}$, that is, $\mathbb{M}^{d}=\mathcal{G}/\mathcal{H}$. Let $\pi \colon \mathcal{G}\to \mathcal{G}/\mathcal{H}$ be the natural projection and $\mathbf{e}$ be the identity element of $\mathcal{G}$. The point $\mathbf{o}=\pi(\mathbf{e})$, which is invariant under the action of the group $\mathcal{H}$, is called the pole of $\mathbb{M}^{d}$. On any manifold $\mathbb{M}^{d}$ under consideration a Riemannian metric $d(\,\cdot\,{,}\,\cdot\,)$, the invariant Haar measure $d\nu$ and an invariant differential operator of the second order $\Delta$, the Laplace–Beltrami operator, are defined. Let $L_{p}=L_{p}(\mathbb{M}^{d},\nu)$ be the set of all $\nu$-measurable functions $\varphi \colon \mathbb{M}^{d}\to\mathbb{R}$ of finite norm $\|\varphi\|_{p}$ given by
$$
\begin{equation*}
\|\varphi \|_{p}= \begin{cases} \displaystyle\biggl(\int_{\mathbb{M}^{d}}|\varphi|^{p}\,d\nu\biggr)^{1/p}, & 1\leqslant p<\infty, \\ \operatorname{ess\,sup} \{|\varphi (x)|\mid x\in \mathbb{M}^{d}\}, & p=\infty. \end{cases}
\end{equation*}
\notag
$$
Now, let $U_{p}=\{\varphi\mid \varphi \in L_{p},\ \|\varphi\|_{p}\leqslant 1\}$. In the local coordinates $x_{l}$, $1\leqslant l\leqslant d$, the Laplace–Beltrami operator $\Delta$ has the form (see [8])
$$
\begin{equation}
\Delta=-(\overline{g})^{-1/2}\sum_{k}\frac{\partial}{\partial x_{k}} \biggl(\sum_{j}g^{jk}(\overline{g})^{1/2} \frac{\partial}{\partial x_{j}}\biggr),
\end{equation}
\tag{2.1}
$$
where
$$
\begin{equation*}
g_{jk}=g\biggl(\frac{\partial}{\partial x_{j}},\frac{\partial}{\partial x_{k}}\biggr)\quad\text{and} \quad \overline{g}=|\det (g_{jk})|, \quad (g^{jk})=(g_{jk})^{-1}.
\end{equation*}
\notag
$$
Let $\mathbb{M}^{d}$ be the sphere $\mathbb{S}^{d}$, $d=2,3, \dots$, the complex projective space $\mathrm{P}^{d}(\mathbb{C})$, $d=4,6,\dots$, the quaternion projective space $\mathrm{P}^{d}(\mathbb{H})$, $d=8,12,\dots$, or the Cayley elliptic plane $\mathrm{P}^{16}(\mathrm{Cay})$. It is well known that the Laplace–Beltrami operator $\Delta$ is elliptic and self-adjoint, and its eigenvalues form a sequence $0\leqslant \theta_{0}\leqslant \theta_{1}\leqslant \cdots \leqslant \theta_{k}\leqslant \dots$ increasing monotonically to infinity. The corresponding eigenspaces $\mathrm{H}_{k}$, $k\geqslant 0$, are finite-dimensional, $d_{k}=\dim \mathrm{H}_{k}<\infty $, $k\geqslant 0$, and orthogonal. Similarly, in the case of the real projective spaces $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$, we have $0\leqslant \theta_{0}\leqslant \theta_{2}\leqslant \cdots \leqslant \theta_{2k}\leqslant \cdots$ and the corresponding eigenspaces are $\mathrm{H}_{2k}$, $k \geqslant 0$. The real Hilbert space $L_{2}$ with the usual inner product has the decomposition $L_{2}=\bigoplus_{k=0}^{\infty}\mathrm{H}_{k}$ ($L_{2}=\bigoplus_{k=0}^{\infty}\mathrm{H}_{2k}$ in the case of $\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$). Let $\{Y_{j}^{k}\}_{j=1}^{d_{k}}$ be some orthonormal basis of $\mathrm{H}_{k}$, $Y_{j}^{k}\colon \mathbb{M}^{d}\to \mathbb{R}$; then for any $\phi \in L_{1}$ we can construct a formal Fourier series:
$$
\begin{equation*}
\phi \sim \sum_{k=0}^{\infty}\sum_{j=1}^{d_{k}}c_{k,j}(\phi) Y_{j}^{k}, \qquad c_{k,j}(\phi)=\int_{\mathbb{M}^{d}}\phi Y_{j}^{k}\,d\nu.
\end{equation*}
\notag
$$
Similarly, in the case of $\mathrm{P}^{d}(\mathbb{R})$, $d=2,3, \dots$, for each $\mathrm{H}_{2k}$ we have an orthonormal basis $\{Y_{j}^{2k}\}_{j=1}^{d_{2k}}$ and for any $\phi \in L_{1}$ we have a formal Fourier series
$$
\begin{equation*}
\phi \sim \sum_{k=0}^{\infty}\sum_{j=1}^{d_{2k}}c_{2k,j}(\phi) Y_{j}^{2k}, \qquad c_{2k,j}(\phi)=\int_{\mathrm{P}^{d}(\mathbb{R})}\phi Y_{j}^{2k}\,d\nu.
\end{equation*}
\notag
$$
All manifolds $\mathbb{M}^{d}$ considered here carry closed geodesics of length $2L$, where $L$ is the diameter of the space $\mathcal{G}/\mathcal{H}$, that is, the maximum distance between two arbitrary points. Recall that a function on $\mathbb{M}^{d}=\mathcal{G}/\mathcal{H}$ is invariant under the left action of the group $\mathcal{H}$ on the manifold $\mathbb{M}^{d}$ if and only if it depends only on the distance between its argument and the pole $\mathbf{o}=\mathbf{e}\mathcal{H}$. A function $Z\colon \mathbb{M}^{d}=\mathcal{G}/\mathcal{H}\to \mathbb{R}$ is called $\mathcal{H}$-invariant or zonal if $Z(\mathrm{h}^{-1}\,\cdot\,)=Z(\,\cdot\,)$ for all $\mathrm{h}\in \mathcal{H}$. Since the distance between any point $x$ on $\mathbb{M}^{d}$ and the pole $\mathbf{o}$ is at most $L$, $\mathcal{H}$-invariant (zonal) functions $Z$ on $\mathbb{M}^{d}$ can be identified with univariate functions $\widetilde{Z}(\cos(2\lambda(\,\cdot\,)))\colon [0,L] \to \mathbb{R}$. More precisely, for each zonal function $Z$ on $\mathbb{M}^{d}$ we have a univariate function $\widetilde{Z}(\,\cdot\,)$, defined by
$$
\begin{equation}
Z(x)=\widetilde{Z}(\cos (2\lambda d(x,\mathbf{o}))), \qquad x\in \mathbb{M}^{d},
\end{equation}
\tag{2.2}
$$
where $\lambda=\pi/(2L)$ for $\mathbb{S}^{d}$, $d=2,3,\dots$, $\mathrm{P}^{d}(\mathbb{C})$, $d=4,6,\dots$, $\mathrm{P}^{d}(\mathbb{H})$, $d=8,12,\dots $, or $\mathrm{P}^{16}(\mathrm{Cay})$. If $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$, then $\lambda=\pi/(4L)$. If $\lambda=\pi/(2L)$ then $\widetilde{Z}(\,\cdot\,)\colon [-1,1] \to \mathbb{R}$, and if $\lambda=\pi/(4L)$, then $\widetilde{Z}(\,\cdot\,)\colon [0,1] \to \mathbb{R}$. Denoting by $\theta$ the distance between an arbitrary point $x\in \mathbb{M}^{d}$ and $\mathbf{o}$ we introduce the geodesic system of the polar coordinates $(\theta,\mathbf{u})$, where $\mathbf{u}$ is the angular parameter. In this coordinate system the representation (2.1) of the Laplace–Beltrami operator splits into two parts: $\Delta=\Delta_{\theta}+\Delta_{\mathbf{u}}$, where $\Delta_{\theta}$ and $\Delta_{\mathbf{u}}$ are the radial and tangential parts of $\Delta$, respectively. The radial part $\Delta_{\theta}$ has the form
$$
\begin{equation}
\Delta_{\theta}=\frac{1}{\mathrm{A}(\theta)}\,\frac{d}{d\theta} \biggl(\mathrm{A}(\theta) \frac{d}{d\theta}\biggr),
\end{equation}
\tag{2.3}
$$
where $\mathrm{A}(\theta)$ is the area of a sphere of radius $\theta$ on $\mathcal{G}/\mathcal{H}$ [7]. The area $\mathrm{A}(\theta)$ can be calculated using the structure of the Lie algebras of $\mathcal{G}$ and $\mathcal{H}$:
$$
\begin{equation}
\mathrm{A}(\theta)=\omega_{\sigma+\rho+1}\lambda^{-\sigma}(2\lambda)^{-\rho} (\sin \lambda \theta)^{\sigma}(\sin 2\lambda \theta)^{\rho},
\end{equation}
\tag{2.4}
$$
where $\omega_{d}$ is the area of the unit sphere in $\mathbb{R}^{d}$ and
$$
\begin{equation}
\begin{gathered} \, \notag \mathbb{S}^{d}\colon \sigma=0, \qquad \rho=d-1, \qquad \lambda=\frac{\pi}{2L}, \qquad d=2,3,\dots, \\ \mathrm{P}^{d}(\mathbb{R}) \colon \sigma=0, \qquad \rho=d-1, \qquad \lambda=\frac{\pi}{4L}, \qquad d=2,3,\dots, \\ \notag \mathrm{P}^{d}(\mathbb{C}) \colon \sigma=d-2, \qquad \rho=1, \qquad \lambda=\frac{\pi}{2L}, \qquad d=4,6,8,\dots, \\ \notag \mathrm{P}^{d}(\mathbb{H}) \colon \sigma=d-4, \qquad \rho=3, \qquad \lambda=\frac{\pi}{2L}, \qquad d=8,12,\dots, \\ \notag \mathrm{P}^{16}(\mathrm{Cay}) \colon \sigma=8, \qquad \rho=7, \qquad \lambda=\frac{\pi}{2L}. \end{gathered}
\end{equation}
\tag{2.5}
$$
Applying (2.3) and (2.4) we see that the radial part of the Laplace–Beltrami operator $\Delta_{\theta }$ can be represented (up to some multiplicative constant) in the form
$$
\begin{equation}
\Delta_{\theta}=\frac{1}{(\sin \lambda \theta)^{\sigma }(\sin 2\lambda \theta)^{\rho}}\frac{d}{d\theta} \biggl((\sin \lambda \theta)^{\sigma}(\sin 2\lambda \theta)^{\rho}\frac{d}{d\theta}\biggr).
\end{equation}
\tag{2.6}
$$
Using a simple change of variables $t=\cos 2\lambda \theta $, we rewrite (2.6) as
$$
\begin{equation}
\Delta_{t}=\frac{1}{(1-t)^{\alpha}(1+t)^{\beta}} \frac{d}{dt}\biggl((1-t)^{1+\alpha}(1+t)^{1+\beta}\frac{d}{dt}\biggr),
\end{equation}
\tag{2.7}
$$
where
$$
\begin{equation}
\alpha=\frac{\sigma+\rho -1}{2} \quad\text{and}\quad \beta=\frac{\rho -1}{2}.
\end{equation}
\tag{2.8}
$$
Observe that for all manifolds $\mathbb{M}^{d}$ under consideration The next statement allows us to express eigenfunctions of $\Delta_{\theta}$ in terms of Jacobi polynomials, which is important for our analysis. Lemma 2.1. The Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ satisfy the linear homogeneous differential equation From Lemma 2.1 we conclude that polynomial eigenfunctions of the operator $-\Delta_{t}$ defined by (2.7) are Jacobi polynomials $P_{k}^{(\alpha,\beta)}$, and eigenvalues are equal to $\theta_{k}=k(k+\alpha +\beta +1)$ (see [7], p. 178, for more details). Consequently, $\mathcal{H}$-invariant or zonal polynomial functions $Z_{k}\in \mathrm{H}_{k}$ on $\mathbb{M}^{d}$, $k\geqslant 1$, $Z_{0}\equiv 1$, can be specified explicitly, since they are eigenfunctions of the Laplace–Beltrami operator.1 Moreover, if $\widetilde{Z}_{k}(\cos (2\lambda (\,\cdot\,)))$ is the function induced on $[0,L]$ by $Z_{k}\in \mathrm{H}_{k}$, $k\geqslant 0$, then
$$
\begin{equation}
\widetilde{Z}_{k}(\theta)=C_{k}(\mathbb{M}^{d}) P_{k}^{(\alpha,\beta)}(\cos 2\lambda \theta),
\end{equation}
\tag{2.10}
$$
where $\alpha$ and $\beta$ were specified above. For example, in the case of the sphere $\mathbb{S}^{d}$, $d=2,3,\dots$, we have $\sigma =0$ and $\rho =d-1$, so and the polynomial $P_{k}^{(\alpha,\beta)}$ reduces to $P_{k}^{((d-2)/2,(d-2)/2)}$ which is a multiple of the Gegenbauer polynomial $P_{k}^{(d-1)/2}$. The addition formula (see, for instance, [8], Theorem 3.2, and [23] and [26]) states that
$$
\begin{equation}
\sum_{j=1}^{d_{k}}Y_{j}^{k}(x) Y_{j}^{k}(y)=\widetilde{Z}_{k}(\theta), \qquad \theta=d(x,y).
\end{equation}
\tag{2.11}
$$
Observe that the normalisation constant $C_{k}(\mathbb{M}^{d})$ in (2.10) should be such that (2.11) is satisfied. We will need explicit formulae for $\dim \mathrm{H}_{k}$ and $\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}$ for our applications. First consider the case when $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. Let $\widetilde{Z}(\cos 2\lambda\theta)$ be the function induced on $[0,L]$ by a zonal function $Z\colon \mathbb{M}^{d} \to \mathbb{R}$ as in (2.2). Since $A(\theta)d\theta$ is the measure induced on $[0,L]$ by the Haar measure $d\nu$ on $\mathbb{M}^{d}$, applying (2.4) we obtain
$$
\begin{equation*}
\begin{aligned} \, I&:=\int_{\mathbb{M}^{d}}Z(x)\, d\nu =\int_{0}^{L}\widetilde{Z}(\cos 2\lambda \theta) A(\theta) \,d\theta \\ &=c_{1}\int_{0}^{L}\widetilde{Z}(\cos 2\lambda \theta) (\sin \lambda \theta)^{\sigma}(\sin 2\lambda \theta)^{\rho}\,d\theta, \end{aligned}
\end{equation*}
\notag
$$
where $c_{1}$ is a positive constant. Making the simple change of variable $t=\cos 2\lambda\theta$ we obtain
$$
\begin{equation}
I=\frac{1}{a}\int_{-1}^{1}\widetilde{Z}(t) (1-t)^{\alpha}(1+t)^{\beta}\,dt,
\end{equation}
\tag{2.12}
$$
where $a$ is a positive constant and $\alpha$ and $\beta$ were defined in (2.8). Put $\widetilde{Z} \equiv 1$ in (2.12). Since $d\nu$ is normalised, we have
$$
\begin{equation*}
1=\int_{\mathbb{M}^{d}}d\nu=\int_{0}^{L}A(\theta)\, d\theta =\frac{1}{a}\int_{-1}^{1}(1-t)^{\alpha}(1+t) ^{\beta}\,dt
\end{equation*}
\notag
$$
or
$$
\begin{equation}
a=\frac{2^{\alpha+\beta+1}\Gamma (\alpha+1) \Gamma (\beta+1)}{\Gamma (\alpha+\beta+2)}.
\end{equation}
\tag{2.13}
$$
We will frequently use formula (2.12) in the text to reduce integration of zonal functions over $\mathbb{M}^{d}$ with respect to $d\nu$ to integration over $[-1,1]$ (or $[0,1]$ in the case of $\mathrm{P}^{d}(\mathbb{R})$) with respect to $(1-t)^{\alpha }(1+t)^{\beta }dt$. Recall that Jacobi polynomials $P_{k}^{(\alpha,\beta)}$, $\alpha >-1$ and $\beta >-1$, are orthogonal on $(-1,1)$ with weight
$$
\begin{equation*}
\omega^{\alpha,\beta}(t)=\frac{(1-t)^{\alpha }(1+t)^{\beta}}{a}, \qquad a>0.
\end{equation*}
\notag
$$
To normalise Jacobi polynomials, we set, as usual,
$$
\begin{equation}
P_{k}^{(\alpha,\beta)}(1) =\frac{\Gamma (\alpha+k+1)}{\Gamma (\alpha+1) \Gamma (k+1)} \asymp k^{\alpha}=k^{(d-2)/2},
\end{equation}
\tag{2.14}
$$
where we used (2.9) in the last equality. This way of normalisation is coming from the definition of Jacobi polynomials in terms of generating functions (see [32], for example). Comparing (2.10) and (2.11) we find that
$$
\begin{equation}
\sum_{j=1}^{d_{k}}Y_{j}^{k}(x) Y_{j}^{k}(y) =C_{k}(\mathbb{M}^{d}) P_{k}^{(\alpha,\beta)}(\cos 2\lambda \theta).
\end{equation}
\tag{2.15}
$$
Setting $y=x$ in (2.15) and integrating with respect to $d\nu $, we obtain
$$
\begin{equation}
d_{k}=\dim \mathrm{H}_{k}=\int_{\mathbb{M}^{d}}\sum_{j=1}^{d_{k}}(Y_{j}^{k}(x))^{2}\,d\nu =C_{k}(\mathbb{M}^{d}) P_{k}^{(\alpha,\beta)}(1).
\end{equation}
\tag{2.16}
$$
Squaring both sides of (2.15) and integrating with respect to the measure $d\nu (x)$ we see that
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\mathbb{M}^{d}}\biggl(\sum_{j=1}^{d_{k}}Y_{j}^{k}(x) Y_{j}^{k}(y)\biggr)^{2}\,d\nu (x) =\sum_{j=1}^{d_{k}}(Y_{j}^{k}(y))^{2} \\ &\qquad =C_{k}^{2}(\mathbb{M}^{d}) \int_{\mathbb{M}^{d}}(P_{k}^{(\alpha,\beta)}(\cos (2\lambda d(x,y))))^{2}\,d\nu (x). \end{aligned}
\end{equation}
\tag{2.17}
$$
Since the measure $d\nu$ is invariant, we find that
$$
\begin{equation*}
\begin{aligned} \, &C_{k}^{2}(\mathbb{M}^{d}) \int_{\mathbb{M}^{d}}(P_{k}^{(\alpha,\beta)}(\cos (2\lambda d(x,y))))^{2}\,d\nu (x) \\ &\qquad =C_{k}^{2}(\mathbb{M}^{d}) \int_{\mathbb{M}^{d}}(P_{k}^{(\alpha,\beta)}(\cos (2\lambda d(x,\mathbf{o}))))^{2}\,d\nu (x)=\frac{\|P_{k}^{(\alpha,\beta)}\|_{2}^{2}}{a}, \end{aligned}
\end{equation*}
\notag
$$
where $a$ is defined by (2.13) and
$$
\begin{equation}
\begin{aligned} \, \notag \|P_{k}^{(\alpha,\beta)}\|_{2}^{2} &:=\int_{-1}^{1}(P_{k}^{(\alpha,\beta)}(t))^{2}(1-t)^{\alpha}(1+t)^{\beta}\,dt \\ &=\frac{2^{\alpha+\beta+1}\Gamma (k+\alpha+1) \Gamma (k+\beta+1)}{(2k+\alpha+\beta+1) \Gamma (k+1) \Gamma (k+\alpha+\beta+1)}\asymp k^{-1} \end{aligned}
\end{equation}
\tag{2.18}
$$
(see [32], formula (4.3.3). Consequently, we can rewrite (2.17) as
$$
\begin{equation*}
\sum_{j=1}^{d_{k}}(Y_{j}^{k}(y))^{2}=\frac{C_{k}^{2}(\mathbb{M}^{d}) \|P_{k}^{(\alpha,\beta)}\|_{2}^{2}}{a},
\end{equation*}
\notag
$$
and integrating the last equality with respect to $d\nu$ we find that
$$
\begin{equation}
d_{k}=\frac{C_{k}^{2}(\mathbb{M}^{d}) \|P_{k}^{(\alpha,\beta)}\|_{2}^{2}}{a}.
\end{equation}
\tag{2.19}
$$
Comparing (2.13), (2.16), (2.19) and (2.9) we see that
$$
\begin{equation}
\begin{aligned} \, \notag d_{k} &=\frac{a(P_{k}^{(\alpha,\beta)}(1))^{2}}{\|P_{k}^{(\alpha,\beta)}\|_{2}^{2}} =\frac{(2k+\alpha+\beta+1) \Gamma (\beta+1) \Gamma (k+\alpha+1) \Gamma (k+\alpha+\beta+1)}{\Gamma (\alpha+1) \Gamma (\alpha+\beta+2) \Gamma (k+1) \Gamma (k+\beta+1)} \\ &\asymp k^{2\alpha+1}=k^{2(d-2)/{2}+1}=k^{d-1}, \qquad k\to \infty. \end{aligned}
\end{equation}
\tag{2.20}
$$
Applying (2.20) we obtain
$$
\begin{equation}
\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}=\sum_{k=0}^{N}d_{k}=a\sum_{k=0}^{N} \frac{(P_{k}^{(\alpha,\beta)}(1)) ^{2}}{\|P_{k}^{(\alpha,\beta)}\|_{2}^{2}}.
\end{equation}
\tag{2.21}
$$
Let us set
$$
\begin{equation*}
K_{N}^{(\alpha,\beta)}(x,y) :=\sum_{k=0}^{N}\frac{ P_{k}^{(\alpha,\beta)}(x) P_{k}^{(\alpha,\beta)}(y)}{\|P_{k}^{(\alpha,\beta)}\|_{2}^{2}};
\end{equation*}
\notag
$$
then (see [32], formula (4.5.3))
$$
\begin{equation}
\begin{aligned} \, \notag K_{N}^{(\alpha,\beta)}(1,1) &=\sum_{k=0}^{N}\frac{(P_{k}^{(\alpha,\beta)}(1))^{2}}{\|P_{k}^{(\alpha,\beta)}\|_{2}^{2}} \\ &=2^{-\alpha -\beta -1}\frac{\Gamma (N+\alpha+\beta+2)}{\Gamma(\alpha+1) \Gamma (N+\beta+1)}P_{N}^{(\alpha+1,\beta)}(1). \end{aligned}
\end{equation}
\tag{2.22}
$$
Comparing (2.21), (2.22) and (2.14) we find that
$$
\begin{equation}
\begin{aligned} \, \notag &\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}=a2^{-\alpha -\beta -1}\frac{\Gamma (N+\alpha+\beta+2)}{\Gamma (\alpha+1) \Gamma (N+\beta+1)}P_{N}^{(\alpha+1,\beta)}(1) \\ &\qquad =\frac{\Gamma (\beta+1) \Gamma (N+\alpha+\beta+2) \Gamma (N+\alpha+2)}{\Gamma (\alpha+\beta+2) \Gamma (\alpha+2) \Gamma (N+\beta+1) \Gamma (N+1)} \asymp N^{2\alpha+2}=N^{d}. \end{aligned}
\end{equation}
\tag{2.23}
$$
In particular, in the case of $\mathbb{S}^{d}$ we have $\alpha=\beta=(d-2)/{2}$ and from (2.23) we obtain
$$
\begin{equation*}
\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}=\frac{(2N+d) (N+d-1)!}{d!\,N!}.
\end{equation*}
\notag
$$
The case of real projective spaces $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$, is more difficult to handle. The spaces $\mathrm{P}^{d}(\mathbb{R})$ can be defined as cosets of the orthogonal group $\mathbf{O}(d+1)$:
$$
\begin{equation*}
\mathrm{P}^{d}(\mathbb{R})=\frac{\mathbf{O}(d+1)}{\mathbf{O}(1) \times \mathbf{O}(d)}.
\end{equation*}
\notag
$$
Consequently, on the real projective spaces $\mathrm{P}^{d}(\mathbb{R})$ only polynomials of even degree appear, because, owing to the identification of antipodal points on
$$
\begin{equation*}
\mathbb{S}^{d}=\frac{\mathbf{O}(d+1)}{\mathbf{O}(d)},
\end{equation*}
\notag
$$
only the even-order polynomials $P_{2k}^{(\alpha,\beta)}$, $k=0,1,2,\dots$, can be lifted to functions on $\mathrm{P}^{d}(\mathbb{R})$. Hence the corresponding eigenspaces of the Laplace–Beltrami operator are $\mathrm{H}_{2k}$, $k\geqslant 0$. Applying (2.5) and (2.8) we obtain $\alpha=\beta=(d-2)/{2}$. Let $Z_{2k}$, $k\in \mathbb{N}$, where $Z_{0}\equiv 1$, be the zonal function corresponding to the eigenvalue and $\widetilde{Z}_{2k}$ be the corresponding function induced on $[0,L]$, $L={\pi}/{2}$. Then $\lambda={\pi}/(4L)={1}/{2}$ and
$$
\begin{equation}
\widetilde{Z}_{2k}(\theta)=C_{2k}(\mathrm{P}^{d}(\mathbb{R})) P_{2k}^{(\alpha,\beta)}(\cos 2\lambda \theta) =C_{2k}(\mathrm{P}^{d}(\mathbb{R})) P_{2k}^{((d-2)/2,(d-2)/2)}(\cos \theta).
\end{equation}
\tag{2.24}
$$
Recall that for $k\in \mathbb{N}$ the polynomial $P_{2k}^{((d-2)/2,(d-2)/2)}$ is just a multiple of the Gegenbauer polynomial $P_{2k}^{((d-1)/2)}$. Applying the well-known formula for the Euler integral of the first kind
$$
\begin{equation*}
\mathrm{B}(x,y)=\int_{0}^{1}\xi^{x-1}(1-\xi)^{y-1}\,d\xi=\frac{\Gamma (x) \Gamma (y)}{\Gamma(x+y)}, \qquad x>0, \quad y>0,
\end{equation*}
\notag
$$
after a simple change of variables we obtain
$$
\begin{equation*}
1=\int_{\mathrm{P}^{d}(\mathbb{R})}\,d\nu =\frac{1}{a^{\ast}}\int_{0}^{1}(1-t^{2})^{(d-2)/2}\,dt =\frac{2^{d-2}(\Gamma (d/2))^{2}}{\Gamma (d)}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
a^{\ast}=\frac{2^{d-2}(\Gamma (d/2))^{2}}{\Gamma (d)}.
\end{equation}
\tag{2.25}
$$
Comparing (2.11) and (2.24) we see that
$$
\begin{equation}
\begin{aligned} \, \notag & \sum_{j=1}^{d_{2k}}Y_{j}^{2k}(x) Y_{j}^{2k}(y)=\widetilde{Z}_{2k}(\theta) \\ &\qquad =C_{2k}(\mathrm{P}^{d}(\mathbb{R})) P_{2k}^{((d-2)/2,(d-2)/2)}(\cos \theta), \qquad \theta=d(x,y). \end{aligned}
\end{equation}
\tag{2.26}
$$
Let $x=y$; then
$$
\begin{equation*}
\sum_{j=1}^{d_{2k}}(Y_{j}^{2k}(x)) ^{2}=C_{2k}(\mathrm{P}^{d}(\mathbb{R}))P_{2k}^{((d-2)/2,(d-2)/2)}(1).
\end{equation*}
\notag
$$
Integrating with respect to $d\nu (x)$ we find that
$$
\begin{equation}
d_{2k}=C_{2k}(\mathrm{P}^{d}(\mathbb{R})) P_{2k}^{((d-2)/2,(d-2)/2)}(1).
\end{equation}
\tag{2.27}
$$
Squaring both sides of (2.26) and integrating with respect to $d\nu (x)$ we obtain
$$
\begin{equation}
\sum_{j=1}^{d_{2k}}(Y_{j}^{2k}(y))^{2}=\frac{C^2_{2k}(\mathrm{P}^{d}(\mathbb{R})) \|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}}{a^{\ast}},
\end{equation}
\tag{2.28}
$$
where $a^{\ast }$ is defined by (2.25) and
$$
\begin{equation}
\bigl\|P_{2k}^{((d-2)/2,(d-2)/2)}\bigr\| _{2\ast}^{2}=\int_{0}^{1}(P_{2k}^{((d-2)/2,(d-2)/2)}(t))^{2}(1-t^{2})^{(d-2)/2}\,dt.
\end{equation}
\tag{2.29}
$$
Since $P_{2k}^{((d-2)/2,(d-2)/2)}(t)$ is an even function, applying (2.18) and (2.29) we see that
$$
\begin{equation}
\bigl\|P_{2k}^{((d-2)/2,(d-2)/2)}\bigr\| _{2\ast}^{2}=\frac{2^{d-2}(\Gamma (2k+d/2) )^{2}}{(4k+d-1) \Gamma (2k+1) \Gamma (2k+d-1)}\asymp k^{-1}.
\end{equation}
\tag{2.30}
$$
Integrating (2.28) with respect to $d\nu (y)$ we find that
$$
\begin{equation}
d_{2k}=\int_{\mathrm{P}^{d}(\mathbb{R})}\sum_{j=1}^{d_{2k}}(Y_{j}^{2k}(y))^{2}\,d\nu (y) =C_{2k}(\mathrm{P}^{d}(\mathbb{R})) \frac{\|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}}{a^{\ast}}.
\end{equation}
\tag{2.31}
$$
From (2.27), (2.31), (2.25) and (2.30) we obtain
$$
\begin{equation}
d_{2k}=\frac{a^{\ast}(P_{2k}^{((d-2)/2,(d-2)/2)}(1))^{2}}{\|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}} =\frac{(4k+d-1) \Gamma (2k+d-1)}{\Gamma (d) \Gamma (2k+1)}\asymp k^{d-1}.
\end{equation}
\tag{2.32}
$$
Finally, to find $\dim \bigoplus_{k=0}^{N}\mathrm{H}_{2k}(\mathrm{P}^{d}(\mathbb{R}))$ we apply [32], formula (4.5.3):
$$
\begin{equation}
\begin{aligned} \, \notag K_{2N}^{((d-2)/2,(d-2)/2)}(x,1) &:=\sum_{k=0}^{2N}\frac{P_{k}^{((d-2)/2,(d-2)/2)}(x) P_{k}^{((d-2)/2,(d-2)/2)}(1)}{\|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}} \\ &=\frac{2^{-d+1}\Gamma (2N+d)}{\Gamma (d/2)\Gamma (2N+d/2)}P_{2N}^{(d/2,(d-2)/2)}(x). \end{aligned}
\end{equation}
\tag{2.33}
$$
Note that (see [32], formula (4.1.3))
$$
\begin{equation}
P_{k}^{(\alpha,\beta)}(x)=(-1)^{k}P_{k}^{(\beta,\alpha)}(-x)
\end{equation}
\tag{2.34}
$$
for all $x\in \mathbb{R}$, $\alpha >-1$, $\beta >-1$ and $k\in \mathbb{N}$, and therefore
$$
\begin{equation*}
\begin{aligned} \, &F(x):=\frac{1}{2}(K_{2N}^{((d-2)/2,(d-2)/2)}(x,1)+K_{2N}^{((d-2)/2,(d-2)/2)}(-x,1)) \\ &=\frac{1}{2}\sum_{k=0}^{2N}\frac{(P_{k}^{((d-2)/2,(d-2)/2)}(x) +(-1)^{k}P_{k}^{((d-2)/2,(d-2)/2)}(x)) P_{k}^{((d-2)/2,(d-2)/2)}(1)}{\|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}} \\ &=\sum_{k=0}^{N}\frac{P_{2k}^{((d-2)/2,(d-2)/2)}(x) P_{2k}^{((d-2)/2,(d-2)/2)}(1)}{\|P_{2k}^{((d-2)/2,(d-2)/2)}\|_{2\ast}^{2}}. \end{aligned}
\end{equation*}
\notag
$$
Consequently, by (2.32),
$$
\begin{equation*}
\dim \bigoplus_{k=0}^{N}\mathrm{H}_{2k}(\mathrm{P}^{d}(\mathbb{R})) =\sum_{k=0}^{N}d_{2k}=a^\ast F(1).
\end{equation*}
\notag
$$
Hence, using (2.33) and (2.34) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\dim \bigoplus_{k=0}^{N}\mathrm{H}_{2k}(\mathrm{P}^{d}(\mathbb{R})) =\frac{\Gamma (d/2) \Gamma (2N+d)}{2\Gamma(d) \Gamma (2N+d/2)}(P_{2N}^{(d/2,(d-2)/2)}(1)+P_{2N}^{(d/2,(d-2)/2)}(-1)) \\ \notag &\qquad=\frac{\Gamma (d/2) \Gamma (2N+d)}{2\Gamma(d) \Gamma (2N+d/2)}(P_{2N}^{(d/2,(d-2)/2)}(1)+P_{2N}^{((d-2)/2,d/2)}(1)) \\ \notag &\qquad =\frac{\Gamma (d/2) \Gamma (2N+d)}{2\Gamma(d) \Gamma (2N+d/2)} \biggl(\frac{\Gamma(2N+d/2+1)}{\Gamma (d/2+1) \Gamma(2N+1)} +\frac{\Gamma (2N+d/2)}{\Gamma(d/2) \Gamma (2N+1)}\biggr) \\ &\qquad =\frac{\Gamma (2N+d) (2N+d)}{d\Gamma (d) \Gamma (2N+1)} =\frac{(2N+d)!}{d!\,(2N)!}\asymp N^{d}. \end{aligned}
\end{equation}
\tag{2.35}
$$
Hence we have proved the following statement. Theorem 2.1. For any $k\geqslant 0$ and $N \geqslant 0$ we have Let us give some comments on this point. The method of the proof of Theorem 2.1 allows us to obtain formulae for $\dim \mathrm{H}_{k}(\mathbb{M}^{d})$ and $\dim\bigoplus_{k=0}^{N}\mathrm{H}_{k}(\mathbb{M}^{d})$ in closed form, where $\mathbb{M}^{d}$ is a two-point homogeneous space. In the case of $\mathbb{S}^{d}$, $d \geqslant 2$, this result is well known for $\dim \mathrm{H}_{k}(\mathbb{S}^{d})$, $k \geqslant 0$. Observe that formula (2.20) was announced in [3] without proof. Here we give a complete proof of (2.20) and find $\dim \mathrm{H}_{k}(\mathbb{M}^{d})$ for all $\mathbb{M}^{d}$ under consideration. The formulae in closed form for $\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}(\mathbb{M}^{d})$ in Theorem 2.1 are new for all manifolds $\mathbb{M}^{d}$ under consideration in this article. There are various approaches to the definition of smoothness via harmonic analysis. We introduce sets of smooth functions by using multiplier operators. This approach allows us to give a unified treatment to a wide range of sets of smooth functions [9]. Consider the case when $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$ first. Let $\varphi$ be an arbitrary function in $L_{p}$, $1\leqslant p\leqslant\infty$, with the formal Fourier expansion
$$
\begin{equation*}
\varphi \sim \sum_{k=0}^{\infty}\sum_{j=1}^{d_{k}}c_{k,j}(\varphi) Y_{j}^{k}
\end{equation*}
\notag
$$
and $\Lambda =\{\lambda (k),\,k\in \{0\} \cup \mathbb{N}\}$ be a sequence of real numbers. If for each $\varphi \in L_{p}$ there is a function $f=\Lambda \varphi \in L_{q}$ such that
$$
\begin{equation}
f\sim \sum_{k=0}^{\infty}\lambda (k) \sum_{j=1}^{d_{k}}c_{k,j}(\varphi) Y_{j}^{k},
\end{equation}
\tag{2.36}
$$
then we say that the multiplier operator $\Lambda$ is of type $(p,q)$. Let $Z\!=\!\widetilde{Z}(\cos (2\lambda d(\mkern-1.5mu\,\cdot\,\mkern-1.5mu,x)))$ be a zonal integrable function on $\mathbb{M}^{d}$. For any $h\in L_{1}$ we define the convolution $g$ by
$$
\begin{equation*}
g(\,\cdot\,)=(Z\ast h) (\,\cdot\,)=\int_{\mathbb{M}^{d}}\widetilde{Z}(\cos (2\lambda d(\,\cdot\,,x))) h(x)\,d\nu (x).
\end{equation*}
\notag
$$
Note that if $Z\in L_{1}$ and
$$
\begin{equation*}
Z\sim \sum_{k=0}^{\infty}\lambda (k) Z_{k}, \qquad Z_{k}\in \mathrm{H}_{k},
\end{equation*}
\notag
$$
then
$$
\begin{equation*}
Z\ast h\sim \sum_{k=0}^{\infty}\lambda (k) \sum_{j=1}^{d_{k}}c_{k,j}(h) Y_{j}^{k}
\end{equation*}
\notag
$$
for each $h\in L_{1}$, that is, the convolution of a function $h\in L_{1}$ with a zonal function $Z\in L_{1}$ acts as the multiplier (diagonal) operator $\Lambda =\{\lambda (k),\,k\in \{0\}\cup\mathbb{N}\}$. For the convolution $g$ we have Young’s inequality
$$
\begin{equation*}
\|Z\ast h\|_{q}\leqslant \|Z\|_{p}\|h\|_{r}, \qquad \frac{1}{q}=\frac{1}{p}+\frac{1}{r}-1, \qquad 1\leqslant p,q,r\leqslant \infty.
\end{equation*}
\notag
$$
Clearly, $\|\Lambda\colon L_{\infty}\to L_{\infty}\|\leqslant\|Z\|_{1}$. We say that $f\in \Lambda U_{p}$ if $\lambda(0)=1$ and $\varphi \in U_{p}$ in (2.36). If $K\in L_{1}$ and then $f = K \ast \varphi$. In particular, Sobolev classes $W_{p}^{\gamma}(\mathbb{M}^{d})$ on $\mathbb{M}^{d}$ are defined as sets of functions $f$ representable in the form
$$
\begin{equation*}
f=I_{\gamma}\ast \varphi\sim c_{0,1}(\varphi)+\sum_{k=1}^{\infty}(k(k+\alpha+\beta+1))^{-\gamma/2} \sum_{j=1}^{d_{k}}c_{k,j}(\varphi) Y_{j}^{k},
\end{equation*}
\notag
$$
where $\|\varphi \|_{p}\leqslant 1$. The Fourier sum $S_{N}(f)$ of order $N$ of $f \in \Lambda U_{p}$ is defined by
$$
\begin{equation*}
S_{N}(f)=\sum_{k=0}^{N}\sum_{j=1}^{d_{k}}c_{k,j}(f) Y_{j}^{k}.
\end{equation*}
\notag
$$
Clearly, if $|\lambda (k)|$ is a nonincreasing sequence, then
$$
\begin{equation*}
\sup_{f\in \Lambda U_{2}}\|f-S_{N-1}(f) \| _{2}=|\lambda (N)|.
\end{equation*}
\notag
$$
In particular, let $\Lambda U_{2}=W^{\gamma}_{2}(\mathbb{M}^{d})$. Then by the definition of linear $n$-widths and (2.23) we have
$$
\begin{equation}
\delta_{n}(W_{2}^{\gamma}(\mathbb{M}^{d}),L_{2}(\mathbb{M}^{d})) \ll n^{-\gamma/d}.
\end{equation}
\tag{2.37}
$$
Similarly, in the case when $\mathbb{M}^{d}=\mathbb{P}^{d}(\mathbb{R})$ let $\varphi \in L_{p}$, $1 \leqslant p \leqslant \infty$,
$$
\begin{equation*}
\varphi \sim \sum_{k=0}^{\infty}\sum_{j=1}^{d_{2k}}c_{2k,j}(\varphi) Y_{j}^{2k},
\end{equation*}
\notag
$$
and let $\Lambda = \{\lambda(k),\,k \in \{0\} \cup \mathbb{N}\}$. We say that $\Lambda$ is of type $(p,q)$ if for each $\varphi \in L_{p}$ there exists a function $f \in L_{q}$ such that
$$
\begin{equation}
f\sim \sum_{k=0}^{\infty}\lambda (k) \sum_{j=1}^{d_{2k}}c_{2k,j}(\varphi) Y_{j}^{2k}.
\end{equation}
\tag{2.38}
$$
We say that $f \in \Lambda U_{p}$ if $\lambda(0)=1$ and $\varphi \in U_{p}$ in (2.38). The Sobolev classes $W^{\gamma}_{p}(\mathbb{P}^{d}(\mathbb{R}))$, $\gamma>0$, are defined as sets of functions representable in the form
$$
\begin{equation*}
f=K\ast \varphi \sim c_{0,1}(\varphi)+\sum_{k=1}^{\infty}(2k(2k+d-1))^{-\gamma/2} \sum_{j=1}^{d_{2k}}c_{2k,j}(\varphi) Y_{j}^{2k},
\end{equation*}
\notag
$$
where $\|\varphi\|_{p} \leqslant 1$. The Fourier sum of order $2N$ is
$$
\begin{equation*}
S_{2N}(f)=\sum_{k=0}^{N}\sum_{j=1}^{d_{2k}}c_{2k,j}(f) Y_{j}^{2k}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\sup_{f\in \Lambda U_{2}}\|f-S_{2(N-1)}(f)\|_{2}=|\lambda (N)|.
\end{equation*}
\notag
$$
Consequently, by the definition of linear $n$-widths and (2.35) we find that
$$
\begin{equation}
\delta_{n}(W_{2}^{\gamma}(\mathrm{P}^{d}(\mathbb{R})), L_{2}(\mathrm{P}^{d}(\mathbb{R}))) \ll n^{-\gamma/d}.
\end{equation}
\tag{2.39}
$$
§ 3. Estimates of entropy numbersTo prove our main results, Theorems 3.1 and 3.3, we need several lemmas. Lemma 3.1. Let $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. Then there exists a multiplier operator $\mathrm{P}$ such that $\mathrm{P}t_{N}=t_{N}$ for all $t_{N}\in \bigoplus_{k=0}^{N}\mathrm{H}_{k}$ and If $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$, then there exists a multiplier $\mathrm{P}$ such that $\mathrm{P}t_{2N}=t_{2N}$ for all $t_{2N}\in \bigoplus_{k=0}^{N}\mathrm{H}_{2k}$ and Proof. We start with the case when $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$, that is, we establish (3.1). Consider the function
$$
\begin{equation}
\mu (x)=1+\int_{0}^{x}\nu (t)\,dt, \qquad x\geqslant 0,
\end{equation}
\tag{3.3}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \nu (t)=\begin{cases} 0, & 0\leqslant t\leqslant \dfrac{1}{2}, \\ \xi (t), & \dfrac{1}{2}<t<1, \\ 0, & t\geqslant 1, \end{cases} \\ \xi (t)=-\omega^{-1}\exp \biggl(\frac{1}{(t-1/2) (t-1)}\biggr) \quad\text{and}\quad \omega=\int_{1/2}^{1}\exp \biggl(\frac{1}{(t-1/2) (t-1)}\biggr)\,dt. \end{gathered}
\end{equation*}
\notag
$$
Let It is easy to check that $\lambda (k,2N)=1$ if $0\leqslant k\leqslant N$ and $\lambda (k,2N)=0$ if $k\geqslant 2N$. Let Applying the Abel transform (summation by parts) in the form
$$
\begin{equation*}
\begin{gathered} \, \sum_{k=0}^{n}u_{k}v_{k}=\sum_{k=0}^{n-1}U_{k}(v_{k}-v_{k+1})+U_{n}v_{n}, \\ U_{k}=\sum_{s=0}^{k}u_{s}, \qquad 0\leqslant k\leqslant n, \end{gathered}
\end{equation*}
\notag
$$
$d+1$ times, where $d$ is the dimension of the manifold under consideration, we obtain
$$
\begin{equation}
K_{2N}=\sum_{k=0}^{2N-d-1}C_{k}^{(d)}S_{k}^{(d)}\Delta^{d+1}\lambda (k,2N) +\sum_{s=0}^{d}C_{2N-s}^{(s)}S_{2N-s}^{(s)}\Delta^{s}\lambda (2N-s,2N)
\end{equation}
\tag{3.5}
$$
and
$$
\begin{equation}
S_{k}^{(\delta)}=\frac{1}{C_{k}^{(\delta)}} \sum_{m=0}^{k}C_{k-m}^{(\delta)}Z_{m},
\end{equation}
\tag{3.6}
$$
where
$$
\begin{equation}
C_{k}^{(\delta)}=\frac{\Gamma (k+\delta+1)}{\Gamma (\delta+1) \Gamma (k+1)}\asymp k^{\delta}, \qquad k\to \infty,
\end{equation}
\tag{3.7}
$$
is the Cesàro kernel, (see, for example, [32], formula (9.4.5)) and the differences $\Delta^{s}\lambda(k,2N)$, $k,s\in \mathbb{N}$, are defined by
$$
\begin{equation*}
\begin{gathered} \, \Delta^{0}\lambda (k,2N)=\lambda (k,2N), \\ \Delta^{1}\lambda (k,2N)=\lambda (k,2N) -\lambda(k+1,2N) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta^{s+1}\lambda (k,2N)=\Delta^{s}\lambda (k,2N) -\Delta^{s}\lambda (k+1,2N).
\end{equation*}
\notag
$$
From (3.5) and (3.7) we find that where
$$
\begin{equation}
\sigma_{1}(N) :=\sum_{k=0}^{2N-d-1}k^{d}|\Delta^{d+1}\lambda (k,2N)|\,\|S_{k}^{(d)}\|_{1},
\end{equation}
\tag{3.9}
$$
and
$$
\begin{equation}
\sigma_{2}(N):=\sum_{s=0}^{d}(2N-s)^{s}|\Delta^{s}\lambda (2N-s,2N)|\,\|S_{2N-s}^{(s)}\|_{1}.
\end{equation}
\tag{3.10}
$$
It was shown in [3], Lemma 3 on p. 316, that
$$
\begin{equation}
\|S_{N}^{(\delta)}\|_{1}\leqslant C \begin{cases} N^{(d-1)/2-\delta}, & 0\leqslant \delta <\dfrac{d-1}{2}, \\ \log N, & \delta=\dfrac{d-1}{2}, \\ 1, & \delta >\dfrac{d-1}{2}. \end{cases}
\end{equation}
\tag{3.11}
$$
Let $\phi\colon \mathbb{R}\to \mathbb{R}$ be a function which has $s$ derivatives. Then for the differences $\Delta^{s}$ of order $s$ we have
$$
\begin{equation}
\begin{aligned} \, \notag |\Delta^{s}\phi (x)| &=\biggl|\sum_{l=0}^{s}(-1)^{l}\frac{s!}{l!\,(s-l)!}\phi(x+l)\biggr| \\ &=\biggl|\int_{0\leqslant t_{l}\leqslant 1,1\leqslant l\leqslant s}\phi^{(s)}\biggl(x+\sum_{l=1}^{s}t_{l}\biggr)\, dt_{1}\cdots dt_{s}\biggr|. \end{aligned}
\end{equation}
\tag{3.12}
$$
Consequently, by (3.12) we have
$$
\begin{equation}
\begin{aligned} \, \notag &|\Delta^{s}\lambda (k,2N)|=\biggl|\Delta^{s}\mu \biggl(\frac{k}{2N}\biggr)\biggr| \\ \notag&\qquad =\biggl|\int_{0\leqslant t_{l}\leqslant (2N)^{-1},1\leqslant l\leqslant s}\mu^{(s)} \biggl(\frac{k}{2N}+\sum_{l=1}^{s}t_{l}\biggr)\,dt_{1}\cdots dt_{s}\biggr| \\ &\qquad \leqslant \int_{0\leqslant t_{l}\leqslant (2N)^{-1},1\leqslant l\leqslant s}\biggl| \mu^{(s)} \biggl(\frac{k}{2N}+\sum_{l=1}^{s}t_{l}\biggr)\biggr|\,dt_{1}\cdots dt_{s}. \end{aligned}
\end{equation}
\tag{3.13}
$$
Observe that $|\mu^{(s)}(x)|=0$, $s\geqslant 1$ if $0\leqslant x\leqslant 1/2$ or $x\geqslant 1$. Moreover,
$$
\begin{equation}
|\mu^{(s)}(x)|\leqslant C(s)\biggl|\biggl(x-\frac{1}{2}\biggr) (x-1)\biggr|^{-2(s-1)}\exp \biggl(\frac{1}{(x-1/2) (x-1)}\biggr)
\end{equation}
\tag{3.14}
$$
for $1/2\leqslant x<1$ and $1 \leqslant s \leqslant d$. Hence it follows from (3.13) that for $0\leqslant k\leqslant 2N$, and therefore from (3.9) and (3.11) we obtain
$$
\begin{equation}
\sigma_{1}(N) \leqslant CN^{-(d+1) }\sum_{k=0}^{2N-d-1}k^{d}\leqslant C.
\end{equation}
\tag{3.15}
$$
Applying (3.14) we find that
$$
\begin{equation}
\begin{aligned} \, \notag |\Delta^{s}\lambda (2N-s,2N)| &\leqslant CN^{s}\exp \biggl(\frac{1}{1-s/(CN)-1}\biggr) \\ &=CN^{s}\exp \biggl(-\frac{CN}{s}\biggr) \leqslant CN^{s}\exp \biggl(-\frac{CN}{d}\biggr) \end{aligned}
\end{equation}
\tag{3.16}
$$
for $1\leqslant s\leqslant d$. Then by (3.10), (3.11) and (3.16). Comparing (3.8), (3.15) and (3.17) we obtain (3.1).
Consider the case when $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$. Put where $\lambda(k,2N)Z_{2k}$ is defined by (3.4). Then the proof proceeds in a similar way. Namely, using the Abel transform we obtain
$$
\begin{equation}
K_{4N}^{\ast} =\sum_{k=0}^{2N-d-1}C_{k}^{(d)}S_{k}^{(d),\ast}\Delta^{d+1}\lambda (k,2N) +\sum_{s=0}^{d}C_{2N-s}^{(s)}S_{2N-s}^{(s)}\Delta^{s}\lambda (2N-s,2N),
\end{equation}
\tag{3.18}
$$
where
$$
\begin{equation*}
S_{k}^{(\delta),\ast}=\frac{1}{C_{k}^{(\delta)}} \sum_{m=0}^{k}C_{k-m}^{(\delta)}Z_{2m}.
\end{equation*}
\notag
$$
From the proof of Theorem 2.1 in [2], p. 230, we obtain
$$
\begin{equation}
\|S_{N}^{(\delta),\ast}\|_{1}\leqslant C \begin{cases} N^{(d-1)/2-\delta}, & 0\leqslant \delta <\dfrac{d-1}{2}, \\ \log N, & \delta=\dfrac{d-1}{2}, \\ 1, & \delta >\dfrac{d-1}{2}. \end{cases}
\end{equation}
\tag{3.19}
$$
Finally, substituting (3.19) into (3.18) and repeating the proof in the case considered above (that is, for $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, or $\mathrm{P}^{16}(\mathrm{Cay})$) we obtain (3.2). The lemma is proved. Let $\mathbf{e}_{1},\dots,\mathbf{e}_{s}$ be the canonical basis in $\mathbb{R}^{s}$, let $\alpha =(\alpha_{1},\dots,\alpha_{s})\in \mathbb{R}^{s}$ and $\beta =(\beta_{1},\dots,\beta_{s})\in \mathbb{R}^{s}$, and let $\langle \alpha,\beta \rangle =\sum_{k=1}^{s}\alpha_{k}\beta_{k}$. Also, let $\|\alpha\|_{(2)}=\langle \alpha,\alpha \rangle^{1/2}$ be the Euclidean norm on $\mathbb{R}^{s}$, let $l_{2}^{s}=(\mathbb{R}^{s},\|\cdot\|_{(2)})$, and let $B_{(2)}^{s}=\{\alpha\in\mathbb{R}^{s}\mid \|\alpha\|_{(2)}\leqslant 1\}$ be the canonical unit ball in $\mathbb{R}^{s}$. The norm $\|\cdot\|_{l_{\infty}^{s}}$ in $l_{\infty}^{s}$ is defined as usual:
$$
\begin{equation*}
\|\alpha \|_{l_{\infty}^{s}}:=\max \{|\alpha_{k}|\mid 1\leqslant k\leqslant s\}.
\end{equation*}
\notag
$$
The unit ball in $l_{\infty }^{s}$ is the cube $Q_{s}=\{\alpha\in\mathbb{R}^{s}\mid |\alpha_{k}|\leqslant 1,\,1\leqslant k\leqslant s\}$. We denote by $\mathrm{Vol}_{s}$ the standard $s $-dimensional volume of subsets of $\mathbb{R}^{s}$. Let $V$ be a convex origin-symmetric (such that $V=-V$) body in $\mathbb{R}^{s}$. Fixing a norm $\|\cdot\|_{V}$ on $\mathbb{R}^{s}$ we denote by $E$ the Banach space $E=(\mathbb{R}^{s},\|\cdot\|_{V})$ with unit ball $V$. For a convex origin-symmetric body $V\subset \mathbb{R}^{s}$ we define the polar body $V^{\mathrm{o}}$ of $V$ by
$$
\begin{equation*}
V^{\mathrm o}=\Bigl\{ \alpha \in \mathbb{R}^{s}\Bigm| \sup_{\beta \in V}|\langle \alpha,\beta \rangle|\leqslant 1\Bigr\}.
\end{equation*}
\notag
$$
The dual space $E^{\mathrm{o}}=(\mathbb{R}^{s},\|\cdot\|_{V^{\mathrm{o}}})$ is endowed with the norm
$$
\begin{equation*}
\|\alpha \|_{V^{\mathrm o}}=\sup \{|\langle\alpha,\beta \rangle|\mid \beta \in V\}.
\end{equation*}
\notag
$$
Let $\{\xi_{k}\mid 1\leqslant k\leqslant s\} \subset L_{\infty}$ be a set of orthonormal functions, let $\Xi_{s}=\mathrm{lin}\{ \xi_{k}\mid {1\leqslant k\leqslant s}\}$, and let $\mathrm{K}$ be the coordinate isomorphism,
$$
\begin{equation*}
\begin{aligned} \, \mathrm{K}\colon \mathbb{R}^{s} & \longrightarrow \Xi_{s}, \\ \alpha & \longmapsto \mathrm{K}\alpha=t^{\alpha}=\sum_{k=1}^{s}\alpha_{k}\xi_{k}. \end{aligned}
\end{equation*}
\notag
$$
The definition $\|\alpha \|_{(\mathrm{K},p)}:=\|t^{\alpha }\|_{p}$ induces a norm on $\mathbb{R}^{s}$. The set
$$
\begin{equation*}
B_{(\mathrm{K},p)}^{s}=\{ \alpha\mid \alpha \in\mathbb{R}^{s},\,\|\alpha \|_{(\mathrm{K} ,p)}\leqslant 1\}, \qquad 1\leqslant p\leqslant \infty,
\end{equation*}
\notag
$$
is a convex origin-symmetric body in $\mathbb{R}^{s}$. Let
$$
\begin{equation*}
r_{k}(\theta)=\mathrm{sign}\sin (2^{k}\pi \theta), \qquad \theta \in (0,1), \quad k\in \mathbb{N},
\end{equation*}
\notag
$$
be the Rademacher functions. We say that a Banach space $X$ with unit ball $U_{X}$ is of Rademacher cotype $2$ if there exists a positive constant $C$ such that for any finite set $\{\varphi_{k}\in X,\,1\leqslant k\leqslant n\}$,
$$
\begin{equation*}
C\int_{0}^{1}d\theta \biggl\|\sum_{k=1}^{n}r_{k}(\theta)\varphi_{k}(\,\cdot\,) \biggr\| \geqslant \biggl(\sum_{k=1}^{n}\|\varphi_{k}(\,\cdot\,) \|^{2}\biggr)^{1/2}.
\end{equation*}
\notag
$$
The smallest such constant $C$ is called the cotype $2$ constant and is denoted by $C_{2}(U_{X})$. Using Khintchine’s inequality
$$
\begin{equation*}
\int_{0}^{1}d\theta\biggl|\sum_{k=1}^{n}r_{k}(\theta)c_{k}\biggr| \geqslant 2^{-1/2}\biggl(\sum_{k=1}^{n}|c_{k}|^{2}\biggr)^{1/2}, \qquad c_{k}\in \mathbb{C}, \qquad 1\leqslant k\leqslant n,
\end{equation*}
\notag
$$
it is easy to show (see, for example, [28], p. 73) that
$$
\begin{equation}
C_2(U_{p}) \leqslant 2^{1/2}, \qquad 1\leqslant p\leqslant 2.
\end{equation}
\tag{3.20}
$$
An analogue of the following statement was proposed by this author in [18]. Lemma 3.2. Let $\{ \xi_{k},\,1\leqslant k\leqslant s\}$ be a set of orthonormal functions on $\mathbb{M}^{d}$ such that $\|\xi_{k}\|_{\infty}\leqslant M$ for any $1\leqslant k\leqslant s$ and some $M>0$. Then Proof. Since $\|M^{-1}\xi_{k}\|_{\infty }\leqslant 1$, $1\leqslant k\leqslant s$, by the orthonormality of the $\xi_{k}$,
$$
\begin{equation*}
\begin{aligned} \, \|\alpha \|_{(\mathrm{K},1)} &=\|K\alpha \|_{1}=\int_{\mathbb{M}^{d}}d\nu (\,\cdot\,)\biggl|\sum_{k=1}^{s}\alpha_{k}\xi_{k}(\,\cdot\,)\biggr| \\ &=\sup_{\|\varphi \|_\infty \leqslant 1}\biggl|\int_{\mathbb{M}^{d}}d\nu (\,\cdot\,) \biggl(\varphi (\,\cdot\,)\sum_{k=1}^{s}\alpha_{k}\xi_{k}(\,\cdot\,)\biggr)\biggr| \\ &\geqslant\biggl|\int_{\mathbb{M}^{d}}d\nu (\,\cdot\,) \biggl(M^{-1}\xi_{j}(\,\cdot\,) \sum_{k=1}^{s}\alpha_{k}\xi_{k}(\,\cdot\,)\biggr)\biggr| =M^{-1}|\alpha_{j}| \end{aligned}
\end{equation*}
\notag
$$
for $1\leqslant j\leqslant s$. This means that The inner John–Löwner (or John) ellipsoid $\mathcal{E}(V)$ associated with a convex body $V\subset \mathbb{R}^{s}$ is the $s$-dimensional ellipsoid of maximum volume contained in $V$. It is well known that for each convex origin-symmetric body $V$ the inner John–Löwner ellipsoid exists and is unique [10], [6]. Let $\mathcal{G}(l_{\infty }^{s})$ be the group of isometries of $l_{\infty}^{s}$. Since $\mathcal{E}(Q_{s})$ is unique and
$$
\begin{equation*}
g(Q_{s})=Q_{s} \quad \text{for all } g\in \mathcal{G}(l_{\infty}^{s}),
\end{equation*}
\notag
$$
we have Thus, from (3.21) and (3.22) it follows that
$$
\begin{equation}
\mathrm{Vol}_{s}(\mathcal{E}(B_{(\mathrm{K},1)}^{s})) \leqslant M^{s}\mathrm{Vol}_{s}(B_{(2)}^{s}).
\end{equation}
\tag{3.23}
$$
Applying Theorem 2 from [4], which states that
$$
\begin{equation*}
\biggl(\frac{\mathrm{Vol}_{s}(V)}{\mathrm{Vol}_{s}(\mathcal{E}(V))}\biggr)^{1/s} \leqslant CC_{2}(V) (\log C_{2}(V))^{4}
\end{equation*}
\notag
$$
for any convex origin-symmetric body $V\subset \mathbb{R}^{s}$, we find from (3.23) that
$$
\begin{equation}
\biggl(\frac{\mathrm{Vol}_{s}(B_{(\mathrm{K},1)}^{s})}{\mathrm{Vol}_{s}(B_{(2)}^{s})}\biggr)^{1/s} \leqslant MCC_{2}(B_{(\mathrm{K},1)}^{s}) (\log C_{2}(B_{(\mathrm{K},1)}^{s}))^{4}.
\end{equation}
\tag{3.24}
$$
Since by the definition of a cotype
$$
\begin{equation*}
C_{2}(B_{(\mathrm{K},1)}^{s})=C_{2}(U_{1}\cap \Xi (s)) \leqslant C_{2}(U_{1}) \leqslant 2^{1/2},
\end{equation*}
\notag
$$
where we have used (3.20), from (3.24) we obtain
$$
\begin{equation}
\biggl(\frac{\mathrm{Vol}_{s}(B_{(\mathrm{K},1)}^{s})}{\mathrm{Vol}_{s}(B_{(2)}^{s})}\biggr)^{1/s}\leqslant MC.
\end{equation}
\tag{3.25}
$$
Comparing (3.25) with the Bourgain–Milman inequality [4] which is valid for any origin-symmetric convex body $V$, we complete the proof. Theorem 3.1. Let $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. Then for any $N\in \mathbb{N}$ and $1\leqslant q\leqslant p\leqslant\infty$ we have where $n=\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}$. If $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$, then where $n=\dim \bigoplus_{k=0}^{N}\mathrm{H}_{2k}$. Proof. We establish the lower bounds in (3.26). It is sufficient to consider only the case $p=\infty$, $q=1$, since all other cases follow by embedding. Let $\mathbb{M}^{d}$ be a two-point homogeneous Riemannian manifold. It is known (see Theorem 2.4 in [30] in the case of $\mathbb{S}^{d}$, $d=2,3,\dots$, $\mathrm{P}^{d}(\mathbb{R})$, $d=2,3,\dots$, $\mathrm{P}^{d}(\mathbb{C})$, $d=4,6,\dots$, or $\mathrm{P}^{d}(\mathbb{H})$, $d=8,12,\dots$, and see [29], p. 3, in the case of $\mathrm{P}^{16}(\mathrm{Cay})$) that for any $\varepsilon >0$ and $N\in \mathbb{N}$ there exists a set of orthonormal functions
$$
\begin{equation*}
\{ \xi_{1},\dots,\xi_{s}\} \subset \bigoplus_{k=0}^{N}\mathrm{H}_{k}
\end{equation*}
\notag
$$
on $\mathbb{M}^{d}$ such that
$$
\begin{equation}
s\geqslant (1-\varepsilon) \dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}=(1-\varepsilon) n
\end{equation}
\tag{3.28}
$$
and $\|\xi_{j}\|_{\infty }\leqslant C(\varepsilon)$, $1\leqslant j\leqslant s$, where $C(\varepsilon)$ depends only on $\varepsilon$. Let $\{x_{1},\dots,x_{N(\delta)}\}$, $N(\delta)\in \mathbb{N}$, $\delta >0$, be a maximal $\delta$-distinguishable net for $(B_{(\mathrm{K},1)}^{s})^{\mathrm{o}}$ in $\|\cdot \|_{(2)}$. Then
$$
\begin{equation}
(B_{(\mathrm{K},1)}^{s})^{\mathrm o}\subset \bigcup_{k=1}^{N(\delta)}(\delta B_{(2)}^{s}+x_{k}).
\end{equation}
\tag{3.29}
$$
Taking volumes in (3.29) we find that
$$
\begin{equation*}
\mathrm{Vol}_{s}((B_{(\mathrm{K},1)}^{s})^{\mathrm o}) \leqslant N(\delta) \delta^{s}\mathrm{Vol}_{s}(B_{(2)}^{s}),
\end{equation*}
\notag
$$
and by Lemma 3.2, where $M=C(\varepsilon)$ depends only on $\varepsilon >0$. Let $\Xi^{\bot}(s)$ denote the orthogonal complement of $\Xi(s)$ in $L_{\infty}$. Then by duality (see, for example, [13])
$$
\begin{equation*}
\begin{aligned} \, &\inf \{ \|\mathrm{K}x_{k}-h_{k}\|_{\infty}\mid h_{k}\in \Xi^{\bot}(s)\} \\ &\qquad =\sup \biggl\{ \int_{\mathbb{M}^{d}}\mathrm{K}x_{k}\xi\, d\nu_{\infty}\biggm|\xi \in \Xi (s) \cap U_{1}\biggr\} =\|x_{k}\|_{(B_{(\mathrm{K},1)}^{s})^{\mathrm o}}\leqslant 1, \end{aligned}
\end{equation*}
\notag
$$
that is, for each $\mathrm{K}x_{k}$ there exists $h_{k}\in \Xi^{\bot}(s)$, $1\leqslant k\leqslant N(\delta)$, such that By Lemma 3.1 there exists a bounded linear multiplier operator $\mathrm{P}$,
$$
\begin{equation*}
\biggl\|\mathrm{P}\biggm| L_{\infty}\to L_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}\biggr\|\ll 1,
\end{equation*}
\notag
$$
which is a projection onto $\oplus_{m=0}^{N}\mathrm{H}_{m}$, that is, $\mathrm{P}x=x$, for all $x\in \bigoplus_{m=0}^{N}\mathrm{H}_{m}$. Since $\mathrm{K}x_{k}\in \bigoplus_{m=0}^{N}\mathrm{H}_{m}$, $1\leqslant k\leqslant N(\delta)$, using (3.31) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\|\mathrm{K}x_{k}-\mathrm{P}h_{k}\|_{\infty} =\|\mathrm{P}(\mathrm{K}x_{k}-h_{k}) \|_{\infty} \\ \notag &\qquad \leqslant \biggl\|\mathrm{P}\biggm|L_{\infty}\to L_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}\biggr\|\,\|\mathrm{K}x_{k}-h_{k}\|_{\infty} \\ &\qquad \leqslant \biggl\|\mathrm{P}\biggm|L_{\infty}\to L_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}\biggr\|\ll 1. \end{aligned}
\end{equation}
\tag{3.32}
$$
Set
$$
\begin{equation*}
g_{k,l}=\frac{(\mathrm{K}x_{k}-\mathrm{P}h_{k}) -(\mathrm{K}x_{l}-\mathrm{P}h_{l})} {2\|P\mid L_{\infty}\to L_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}\|}, \qquad 1\leqslant k,l\leqslant N(\delta).
\end{equation*}
\notag
$$
Using (3.32) we obtain Recall that $\mathrm{P}$ is a multiplier. Hence $\mathrm{P}$ is self-adjoint, $\mathrm{P}^{\ast }=\mathrm{P}$, and therefore
$$
\begin{equation*}
\begin{aligned} \, &[ \mathrm{K}x_{k}-\mathrm{K}x_{l},\mathrm{P}h_{k}-\mathrm{P}h_{l}] =\int_{\mathbb{M}^{d}}(\mathrm{K}x_{k}-\mathrm{K}x_{l}) (\mathrm{P}h_{k}-\mathrm{P}h_{l}) \,d\nu \\ &\qquad =\int_{\mathbb{M}^{d}}(\mathrm{P}^{\ast}\mathrm{K}x_{k}-\mathrm{P}^{\ast}\mathrm{K}x_{l}) (h_{k}-h_{l})\, d\nu =\int_{\mathbb{M}^{d}}(\mathrm{K}x_{k}-\mathrm{K}x_{l}) (h_{k}-h_{l}) \,d\nu=0 \end{aligned}
\end{equation*}
\notag
$$
for any $1\leqslant k\neq l\leqslant N(\delta)$, since $\mathrm{K}x_{k}\in \Xi (s)\subset \bigoplus_{m=0}^{N}\mathrm{H}_{m}$ and $h_{k}\in \Xi^{\bot}(s)$. Consequently, applying (3.33) we see that
$$
\begin{equation*}
\begin{aligned} \, \delta^{2} &\leqslant \|x_{k}-x_{l}\|_{(2) }^{2}=\|\mathrm{K}x_{k}-\mathrm{K}x_{l}\|_{2}^{2} \leqslant \|\mathrm{K}x_{k}-\mathrm{K}x_{l}\| _{2}^{2}+\|\mathrm{P}h_{k}-\mathrm{P}h_{l}\|_{2}^{2} \\ &=\|(\mathrm{K}x_{k}-\mathrm{P}h_{k}) -(\mathrm{K} x_{l}-\mathrm{P}h_{l}) \|_{2}^{2}\leqslant C\|g_{k,l}\|_{2}^{2} \\ &=C\int_{\mathbb{M}^{d}}g_{k,l}^{2}\,d\nu \leqslant C\|g_{k,l}\|_{1}\,\|g_{k,l}\|_{\infty} \leqslant C\|g_{k,l}\|_{1}. \end{aligned}
\end{equation*}
\notag
$$
Hence, there are
$$
\begin{equation*}
\vartheta_{k}:=C(\mathrm{K}x_{k}-\mathrm{P}h_{k}), \qquad 1\leqslant k,l\leqslant N(\delta),
\end{equation*}
\notag
$$
that form a $C\delta^{2}$-distinguishable net in $U_{\infty }\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}$. Observe that $N(\delta)$ is bounded for any $\delta >0$ (since $U_{\infty }\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}$ is finite dimensional and bounded in $L_{1}$) and increasing as $\delta\to0$. Given $n \in \mathbb{N}$, let $\delta$ be the smallest number such that $N(\delta) \leqslant 2^{n}$. Then applying (3.30) we find that or In particular, let $n=\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}$. Then by the definition of entropy numbers, for any $N \in \mathbb{N}$ we have
$$
\begin{equation}
e_{n}\biggl(U_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m},L_{1}\biggr) \geqslant C\delta^{2}.
\end{equation}
\tag{3.35}
$$
Recall that $s \asymp n$ by (3.28). Consequently, applying (3.35) and (3.34) we obtain Clearly,
To show (3.27), that is, in the case when $\mathbb{M}^{d}=\mathrm{P}^{d}(\mathbb{R})$ we must apply (3.2) and just repeat the proof of (3.26). Theorem 3.1 is proved. We will need the following statement (Theorem 2 in [3], p.317). Theorem 3.2. Let $\Lambda=\{\lambda(k),\,k\in\{0\}\cup\mathbb{N}\}$ be a multiplier operator, $\Lambda \colon L_{p} \to L_{q}$, $1 \leqslant q \leqslant p \leqslant \infty$, and let Observe that the proof of Theorem 3.2 in the case when $\mathbb{M}^{d}=P^{d}(\mathbb{M}^{d})$ is the same as in the case when $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$ and uses estimates (3.19) for the $L_{1}$-norms of Cesàro kernels. The next statement gives us estimates for the entropy of multiplier operators in terms of $\{\lambda(k),\,k\in\{0\}\cup\mathbb{N}\}$. Theorem 3.3. Let $\Lambda =\{\lambda(k),\,k \in \{0\} \cup \mathbb{N}\}$ be such that $\lambda (k)\neq 0$ for $k\geqslant 0$, Proof. Consider the case when $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. The proof in the case of $\mathrm{P}^{d}(\mathbb{R})$ is similar. We start with the upper bounds. It was shown in [3], p. 318, that if $\Lambda$ satisfies conditions (3.36) and (3.37), then
$$
\begin{equation*}
\sup_{f\in \Lambda U_{p}}\|f-t_{N}(f) \|_{q}\ll \sum_{k=N+1}^{\infty}|\Delta^{l+1}\lambda (k)|k^{l}=u_{N}, \qquad 1\leqslant q\leqslant \infty,
\end{equation*}
\notag
$$
where $l$ is defined by (3.36),
$$
\begin{equation*}
t_{N}(f)=t_{N}(\Lambda \ast \varphi) =\lambda (0) c_{0,1}(\varphi) +\biggl(\sum_{k=N+1}^{\infty}\Delta^{l+1}\lambda (k)C_{k}^{(l)}S_{k}^{(l)}\biggr) \ast \varphi
\end{equation*}
\notag
$$
and $C_{k}^{l}$ and $S_{k}^{l}$ are defined by (3.7) and (3.6), respectively. Applying Theorem 3.2 we obtain
$$
\begin{equation*}
\|\Lambda\mid L_{p}\longrightarrow L_{q}\|<\infty, \qquad 1\leqslant q\leqslant p\leqslant \infty.
\end{equation*}
\notag
$$
Also, since the series in (3.36) is convergent, it follows that This implies that $\Lambda \colon L_{p}\longrightarrow L_{q}$ is compact. If $1\leqslant q\leqslant p\leqslant \infty$, then by embedding and the definition of linear widths we have
$$
\begin{equation*}
\delta_{n}(\Lambda U_{p},L_{q}) \leqslant \delta_{n}(\Lambda U_{q},L_{q}) \ll u_{N}, \qquad 1\leqslant q\leqslant \infty,
\end{equation*}
\notag
$$
for any $n$ such that
$$
\begin{equation*}
\dim \bigoplus_{m=0}^{N-1}\mathrm{H}_{m}\leqslant n\leqslant \dim \bigoplus_{m=0}^{N} \mathrm{H}_{m}-1, \qquad N\in \mathbb{N}.
\end{equation*}
\notag
$$
Since the $u_{N}$ form a nonincreasing sequence, the $u_{N}^{-1}$ are nondecreasing. Hence $f(n)$ in (1.2) is nondecreasing and satisfies condition (1.1). Applying (1.2) we obtain
Let us turn to lower bounds. Let $t_{N}\in \bigoplus_{m=0}^{N}\mathrm{H}_{m}$ be any polynomial, and let
$$
\begin{equation*}
\mathrm{M}=\biggl\{ \mu \biggl(\frac{k}{2N}\biggr),\ k\in \{0\} \cup \mathbb{N}\biggr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Lambda^{-1}=\{ \lambda^{-1}(k),\ k\in \{0\} \cup \mathbb{N}\} \quad\text{and}\quad \mathrm{H}=\{ \eta (k),\ k\in \{ 0\} \cup\mathbb{N}\}.
\end{equation*}
\notag
$$
Since $\mu (k/(2N))=1$ if $0 \leqslant k\leqslant N$ and $\mu (k/(2N))=0$ if $k\geqslant 2N$, applying Theorem 3.2 we obtain
$$
\begin{equation*}
\|\Lambda^{-1}t_{N}\|_{\infty}=\|\Lambda^{-1}\mathrm{M}t_{N}\|_{\infty} =\|\mathrm{H}t_{N}\|_{\infty} \ll l_{N}\|t_{N}\|_{\infty}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\|\Lambda^{-1}t_{2N}\|_{\infty}\ll l_{2N}\| t_{2N}\|_{\infty},
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
C l_{2N}^{-1}U_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m}\subset \Lambda U_{\infty}.
\end{equation*}
\notag
$$
Thus, by an embedding and Theorem 3.1,
$$
\begin{equation}
e_{n}(\Lambda U_{p},L_{q}) \gg e_{n}(\Lambda U_{\infty},L_{1}) \gg l_{2N}^{-1}e_{n}\biggl(U_{\infty}\cap \bigoplus_{m=0}^{2N}\mathrm{H}_{m},L_{1}\biggr) \gg l_{2N}^{-1}.
\end{equation}
\tag{3.40}
$$
Comparing (3.39) and (3.40) we obtain the proof of Theorem 3.3. Theorem 3.3 gives sharp order estimates in various important cases, for example, in the case of Sobolev classes. Corollary 3.1. Let $\Lambda =\{\lambda (k),\,k\in \{0\} \cup \mathbb{N}\} $, where Proof. Let $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. Observe that for $t>0$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \lambda^{(s)}(t) &=((t(t+\alpha+\beta+1))^{-\gamma/2})^{(s)} \\ &=\sum_{k=0}^{s}\frac{s!}{k!\,(s-k)!}(t^{-\gamma/2})^{(k)}((t+\alpha+\beta+1)^{-\gamma/2})^{(s-k)} \asymp (-1)^{s}t^{-\gamma -s}. \end{aligned}
\end{equation}
\tag{3.41}
$$
Hence, applying (3.12), for $k \geqslant 1$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag |\Delta^{s}\lambda (k)| &=\biggl|\sum_{m=0}^{s}(-1)^{m}\frac{s!}{m!\,(s-m)!}\lambda(k+m)\biggr| \\ &=\biggl|\int_{0\leqslant t_{m}\leqslant 1,1\leqslant m\leqslant s}\lambda^{(s)}\biggl(k+\sum_{m=1}^{s}t_{m}\biggr)\, dt_{1}\cdots dt_{s}\biggr| \asymp k^{-\gamma -s}, \end{aligned}
\end{equation}
\tag{3.42}
$$
where we used (3.41) in the last relation. Thus, by (3.42)
$$
\begin{equation}
u_{N}=\sum_{k=N+1}^{\infty}|\Delta^{l+1}\lambda (k)|k^{l}\asymp \sum_{k=N+1}^{\infty}k^{-\gamma -l-1}k^{l}\asymp N^{-\gamma}.
\end{equation}
\tag{3.43}
$$
Also, since $\gamma >0$, we have
$$
\begin{equation*}
\lim_{k\to \infty}|\Delta^{s}\lambda (k)|k^{s}\ll \lim_{k\to \infty}k^{-\gamma -s}k^{s}=0
\end{equation*}
\notag
$$
for any $0 \leqslant s \leqslant l$. Hence (3.36) and (3.37) are satisfied. By Theorem 2.1 or $N\asymp n^{{1}/{d}}$. Applying (3.43) we obtain Therefore, $f(n) \asymp n^{\gamma/d}$ and, in particular,
$$
\begin{equation*}
f(2^{j}) \asymp 2^{{\gamma j}/{d}}=2^{\gamma/d}2^{\gamma (j-1)/d}\asymp f(2^{j-1}).
\end{equation*}
\notag
$$
Consequently, $f(n)$ defined by (3.38) satisfies (1.1).
To estimate $l_{N}$ we notice that
$$
\begin{equation*}
\eta (t)=(t(t+\alpha+\beta+1))^{\gamma/2}\mu \biggl(\frac{t}{2N}\biggr), \qquad t\geqslant 0.
\end{equation*}
\notag
$$
Applying (3.41) we obtain
$$
\begin{equation*}
\begin{aligned} \, |\eta^{(l+1)}(t)| &=\biggl|\sum_{m=0}^{l+1}\frac{(l+1)!}{m!\,(l+1-m)!} ((t(t+\alpha+\beta+1))^{{\gamma}/{2}})^{(m)}\mu^{(l+1-m)}\biggl(\frac{t}{2N}\biggr)\biggr| \\ &\ll C(l) \sum_{m=0}^{l+1}t^{\gamma -m}N^{-l-1+m}, \end{aligned}
\end{equation*}
\notag
$$
since $|\mu^{(s)}(t)|\leqslant C(s)$, $1 \leqslant s \leqslant l+1$. Hence, for $k \geqslant 1$ we find that
$$
\begin{equation*}
\begin{aligned} \, |\Delta^{l+1}\eta(k)| &=\biggl|\int_{0\leqslant t_{s}\leqslant 1,1\leqslant s\leqslant l+1}\eta^{(l+1)} \biggl(k+\sum_{s=1}^{l+1}t_{s}\biggr)\, dt_{1}\cdots dt_{l+1}\biggr| \\ &\ll\biggl|\int_{0\leqslant t_{s}\leqslant 1,1\leqslant s\leqslant l+1}\sum_{m=0}^{l+1}N^{-l-1+m} \biggl(k+\sum_{s=1}^{l+1}t_{s}\biggr)^{\gamma-m}\,dt_{1}\cdots dt_{l+1}\biggr| \\ &\ll \sum_{m=0}^{l+1}N^{-l-1+m}k^{\gamma -m}. \end{aligned}
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation}
\begin{aligned} \, \notag l_{N} &=|\eta (0)|+\sum_{k=1}^{2N}| \Delta^{l+1}\eta (k)|k^{l} \ll \sum_{k=1}^{2N}k^{l}\sum_{m=0}^{l+1}N^{-l-1+m}k^{\gamma -m} \\ &=\sum_{m=0}^{l+1}N^{-l-1+m}\sum_{k=1}^{2N}k^{l}k^{\gamma -m} \ll \sum_{m=0}^{l+1}N^{-l-1+m}N^{l+\gamma -m+1}\ll N^{\gamma}, \end{aligned}
\end{equation}
\tag{3.44}
$$
since $l$ is defined by (3.36) and $d$ is fixed. Applying Theorem 3.3, (3.43) and (3.44) we obtain
$$
\begin{equation*}
e_{n}(W_{p}^{\gamma},L_{q}) \asymp N^{-\gamma}, \qquad 1\leqslant q\leqslant p\leqslant \infty.
\end{equation*}
\notag
$$
By Theorem 2.1, $n=\dim \bigoplus_{k=0}^{N}\mathrm{H}_{k}\asymp N^{d}$ for any $\mathbb{M}^{d}$ under consideration. Thus, applying Theorem 3.3 we obtain the proof for $\mathbb{M}^{d}=\mathbb{S}^{d}$, $\mathrm{P}^{d}(\mathbb{C})$, $\mathrm{P}^{d}(\mathbb{H})$, $\mathrm{P}^{16}(\mathrm{Cay})$. In the case of $\mathrm{P}^{d}(\mathbb{R})$ the proof is similar.
Citation in format AMSBIB
\Bibitem{Kus25} |
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