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Supersmooth tile T. I. Zaitsevaab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
Bibliographic databases:
    § 1. IntroductionA cardinal $\mathrm B$-spline is a self-convolution of the characteristic function of the interval $[0, 1]$,
$$
\begin{equation*}
B_{\ell}=\underbrace{\chi_{[0,1]} * \dots * \chi_{[0,1]}}_{\ell+1},
\end{equation*}
\notag
$$
here $\ell$ is the order of the $\mathrm B$-spline in question. In particular, $B_0=\chi_{[0, 1]}$, $B_1=\chi_{[0, 1]} * \chi_{[0, 1]}$ and so on. Any $\mathrm B$-spline $B_{\ell}$ has knots at integer points and lies in the space $C^{\ell-1}(\mathbb R)$. $\mathrm B$-splines are widely useful in approximation theory. Applications of $\mathrm B$-splines to the construction of orthogonal wavelets, and also to subdivision algorithms of iterative modelling of surfaces, are based on the fact that they satisfy refinement equations. Definition 1. A refinable function $\varphi(x)$ with dilation coefficient $2$ is a solution of the scalar refinement equation Taking the Fourier transform we obtain
$$
\begin{equation*}
\widehat{\varphi}(2 \xi)=a(\xi) \widehat{\varphi}(\xi),
\end{equation*}
\notag
$$
where the trigonometric polynomial $a(\xi)=\frac{1}{2} \sum_{k=0}^{N} c_{k} e^{-2 \pi i k \xi}$ is the mask of this equation. From properties of the Fourier transform it follows that if two functions $\varphi_1$ and $\varphi_2$ satisfy the refinement equation with masks $a_1(\xi)$ and $a_2(\xi)$, then their convolution also satisfies the refinement equation with mask $a_1(\xi)a_2(\xi)$. The characteristic function of $[0, 1]$ is a solution of the refinement equation with mask $a_0(\xi)=(1+e^{-2\pi i \xi}) / 2$. Therefore, any cardinal $\mathrm B$-spline $B_\ell$ also satisfies the refinement equation with mask $a_\ell(\xi)=2^{-(\ell+1)}(1+e^{-2\pi i \xi})^{\ell+1}$. This equation has the coefficients $c_0=2^{-\ell}\binom{\ell+1}{0}$, $c_1=2^{-\ell}\binom{\ell+1}{1}$, $\dots$ and $c_{\ell+ 1}=2^{-\ell}\binom{\ell+1}{\ell+1}$, where the $\binom{\ell+1}{k}$ are binomial coefficients. By a multivariate $\mathrm B$-spline one usually means the direct product of some one-dimensional $\mathrm B$-splines, that is,
$$
\begin{equation*}
B_\ell(\boldsymbol{x})=B_\ell(x_1, \dots, x_n)=B_\ell(x_1) \cdots B_\ell(x_n).
\end{equation*}
\notag
$$
These classical multivariate $\mathrm B$-splines also satisfy refinement equations with mask $\boldsymbol{a}_\ell(\xi_1, \dots, \xi_n)=a_\ell(\xi_1) \cdots a_\ell(\xi_n)$. More general tile $\mathrm B$-splines were introduced in [7], where it was also shown that they have some advantages over the classical $\mathrm B$-splines. We will consider these advantages below. A similar (but less general) construction was studied in [49] and [51]. Let us see how tile $\mathrm B$-spline are constructed. The concept underlying the construction is a tile, a special self-affine compact set (see Definition 4). Definition 2. A tile $\mathrm B$-spline constructed from a tile $G$ is the self-convolution of several characteristic functions of this tile, Definition 3. A multivariate refinement equation is an equation of the form where $M$ is an expanding integer matrix, that is, all of its eigenvalues lie outside the unit disc. Solutions of a refinement equation are refinable functions. An application of the Fourier transform shows that
$$
\begin{equation}
\widehat{\varphi}(M^\top\boldsymbol{\xi})=\boldsymbol{a}(\boldsymbol{\xi}) \widehat{\varphi}(\boldsymbol{\xi}),
\end{equation}
\tag{1.2}
$$
where $\boldsymbol{a}(\boldsymbol{\xi})=\frac{1}{m} \sum_{\boldsymbol{k} \in \mathbb Z^n} c_{\boldsymbol{k}} e^{-2 \pi i (\boldsymbol{k}, \boldsymbol{\xi})}$ is the mask of the refinement equation, ${m=|{\det{M}}|}$. The characteristic function of the tile $G$ satisfies the refinement equation (2.1) with mask $a_0$ defined by (2.2). For any tile $\mathrm B$-spline $B_{\ell}^G$ we have the mask $a=a_0^\ell$. This is proved as in the scalar case, by using the properties of the Fourier transform. So tile $\mathrm B$-splines are solutions of refinement equations. Refinement equations have extensively been studied (see [3], [6], [10], [14] and [43]) in relation to approximation theory, graphic design, random power series, combinatorial theory of numbers and discrete geometry (see also [24], [27], [28], [37], [39] and [44]). Usually, a solution of a refinement equation cannot be expressed explicitly in terms of elementary functions; however, there are fast algorithms for calculating it with prescribed accuracy. In addition, certain characteristics of the solution can explicitly be found from the refinement equation, for example, its smoothness in the spaces $C$ and $L_2$. Partial derivatives of the solution, which also satisfy the refinement equations, can be found using the same machinery. In the present paper we are concerned with the evaluation of the smoothness of tile $\mathrm B$-splines in $L_2$. Surprisingly, some tile $\mathrm B$–splines feature greater smoothness than the classical $\mathrm B$-splines of the same order [7], [50]. Such splines and the corresponding tiles will be called supersmooth. In applications to geometrical modelling (subdivision schemes) the smoothness of tile $\mathrm B$-splines coincides with that of the surfaces generated, and so smoother splines result in smoother surfaces. There are no supersmooth tiles in the one-dimensional case; in this case standard cardinal $\mathrm B$-splines are the most smooth ones among the refinable functions with given size of support [3]. However, as we will see, there are sufficiently many supersmooth families in the two-dimensional case. Tiles are defined in § 2; a brief survey of the available methods for the evaluation of smoothness of refinable functions is given in § 3. In § 4 we introduce some auxiliary objects, namely, tile cylinders and rational spaces, and establish their basic properties; in § 5 we apply these properties to the proof of a theorem on the smoothness of tile $\mathrm B$-splines. In the last section, § 6, we prove some results on supersmooth tile $\mathrm B$-splines, give illustrative examples, and present a large number of new families of supersmooth tiles; we also formulate a conjecture on the complete classification of such tiles in the case of few digits. § 2. Main definitionsConsider an expanding integer matrix $M \in \mathbb Z^{n \times n}$, that is, a matrix with all eigenvalues lying outside the unit disc. In what follows this matrix define the linear part of the affine dilation used in the construction of self-affine tiles. The matrix $M \in \mathbb Z^{n \times n}$ defines a partition of the lattice $\mathbb Z^n$ into $m=|{\det M}|$ equivalence classes:
$$
\begin{equation*}
\boldsymbol{y} \sim \boldsymbol{x} \quad \Longleftrightarrow\quad \boldsymbol{y}-\boldsymbol{x} \in M\mathbb Z^n.
\end{equation*}
\notag
$$
We choose one representative $\boldsymbol{d}_i \in \mathbb Z^n$ of each equivalence class; the resulting set is called a set of digits:
$$
\begin{equation*}
D(M)=\bigl\{\boldsymbol{d}_i \colon i=0, \dots, m-1\bigr\}.
\end{equation*}
\notag
$$
We will always assume that $\mathbf 0 \in D(M)$. There are infinitely many sets of digits corresponding to a single matrix. If $M$ is a scalar, then $D(M)$ is a set of digits in the number system with base $m$. Thus, an integer matrix and a set of digits define a ‘number system’ in $\mathbb Z^n$. The set
$$
\begin{equation*}
G=\biggl\{ \sum _{k=1}^{\infty}M^{-k}\boldsymbol{\Delta}_k \biggm| \boldsymbol{\Delta}_k \in D(M)\biggr\}
\end{equation*}
\notag
$$
is an analogue of the unit interval in this system. It is known (see [28]) that $G$ is compact and self-affine:
$$
\begin{equation*}
G=\bigcup_{\boldsymbol{\Delta} \in D(M)} {M^{-1}(G+\boldsymbol{\Delta})},
\end{equation*}
\notag
$$
and any two different sets $M^{-1}(G+\boldsymbol{\Delta})$ intersect in a nullset. It is also known that the Lebesgue measure of $G$ is a positive integer. We consider only the case where $G$ has Lebesgue measure 1 (in this case one can use tiles for the construction of multivariate Haar systems [39]). Definition 4. A tile generated by an expanding matrix $M \in \mathbb Z^{n\times n}$ and a set of digits $D(M)=\{\boldsymbol{d}_i \colon i=0, \dots, m-1\}$ is the set provided that the Lebesgue measure of $G$ is 1. The integer translates $\{G+\boldsymbol{k} \mid \boldsymbol{k} \in \mathbb Z^n\}$ of $G$ tile the whole plane $\mathbb R^n$ with one layer — this explains the term ‘tile’. Another name for a tile is an ‘integral self-affine set with standard digit set’. By self-affinity the characteristic function $\varphi=\chi_G$ of $G$ satisfies, almost everywhere on $\mathbb R^n$, the refinement equation
$$
\begin{equation}
\varphi(\boldsymbol{x})={\sum _{\boldsymbol{\Delta} \in D(M)}{\varphi(M\boldsymbol{x}- \boldsymbol{\Delta})}}, \qquad \boldsymbol{x} \in \mathbb R^n.
\end{equation}
\tag{2.1}
$$
In this case $c_{\boldsymbol{k}}=1$ for all $\boldsymbol{k} \in D(M)$ and $c_{\boldsymbol{k}}= 0$ for $\boldsymbol{k} \notin D(M)$, and the mask is
$$
\begin{equation}
a_0(\boldsymbol{\xi})=\frac{1}{m} \sum _{\boldsymbol{\Delta} \in D(M)} e^{-2\pi i (\boldsymbol{\Delta}, \boldsymbol{\xi})}.
\end{equation}
\tag{2.2}
$$
The number of digits is important, because it controls the number of coefficients in the refinement equation for tiles and tile $\mathrm B$-splines. Example 1. In the case where $d=1$, for $M=3$ we can take $D(M)=\{0, 1, 2\}$. Then The characteristic function $\varphi = \chi_{[0, 1]}$ satisfies the refinement equation $\varphi(x)=\varphi(3x)+\varphi(3x-1)+\varphi(3x-2)$. If $D(M)=\{0, 1, 5\}$, then since $x \leqslant 5 \sum_{k=1}^{\infty} 3^{-k}=2.5$ for each $x \in G$. Example 2. Consider the matrix $M=\begin{pmatrix}3 & 0 \\ 0 & 3\end{pmatrix}$; then $m= |{\det M}|=9$ (this is an example from [31]). As a set of digits, we can take Example 2 will be considered in § 5 to illustrate the concept of ‘tile cylinders’. The isotropic case, where a tile is generated by an isotropic matrix $M$, will be important for us. This means that the eigenvalues of the matrix are equal in absolute value, and the matrix has no nontrivial Jordan blocks. So an isotropic matrix is similar to an orthogonal matrix times a scalar. The isotropic case appears frequently in the literature because of its simplicity and similarity to the scalar setting. We need some notation. We let $\rho(A)$ denote the spectral radius of the matrix $A$, and $r=\rho(M)$ denote the spectral radius of the matrix $M$ defining the dilation in the refinement equation. Let $\mathcal{S}$ be the Schwartz space of smooth rapidly decreasing functions, $\mathcal{S}'$ be the space of distributions on $\mathcal{S}$ and $\mathcal{S}_0'$ be the set of distributions in $\mathcal{S}'$ with compact support. We denote vectors by boldface characters, $\boldsymbol{x}=( x_1, \dots, x_n) \in\mathbb R^n$, and scalars are denoted in lightface. The Fourier transform in $\mathbb R^n$ is defined by
$$
\begin{equation*}
\widehat{f} (\boldsymbol{\xi})= \int_{\mathbb R^n}f(\boldsymbol{x}) e^{-2\pi i (\boldsymbol{x}, \boldsymbol{\xi})}d\boldsymbol{x}.
\end{equation*}
\notag
$$
Let ${\mathbb T}_n$ denote the interval of periodicity $[0, 1]^n$. We consider trigonometric polynomials of the form
$$
\begin{equation*}
\boldsymbol{p}(\boldsymbol{\xi})= \sum_{\boldsymbol{k} \in \Omega} c_{\boldsymbol{k}}e^{-2\pi i (\boldsymbol{k}, \boldsymbol{\xi})}, \qquad \boldsymbol{\xi} \in \mathbb R^n,
\end{equation*}
\notag
$$
where $\Omega$ is a finite subset of $\mathbb{Z}^n$. The set of all such polynomials is denoted by $\mathcal{P}_{\Omega}$. The spectrum of the polynomial $\boldsymbol{p}(\boldsymbol{\xi})$ is the set $\{\boldsymbol{k} \mid c_{\boldsymbol{k}} \ne 0\}$.
§ 3. Methods of evaluation of the smoothness of refinement equationsIn this section we discuss briefly the possible approaches to the evaluation of the smoothness of general refinement equations, and then we obtain applications to tile $\mathrm B$-splines. We consider only equations with a finite number of terms which satisfy the standard assumption $\sum_{\boldsymbol{k} \in \mathbb Z^n}c_{\boldsymbol{k}}=m$, where $m=|{\det M}|$. This condition is satisfied by tile $\mathrm B$-splines. In this case the refinement equation has a unique (up to a multiplicative constant) solution $\varphi$ in the space $\mathcal{S}_0'(\mathbb R^n)$. We normalize the refinable function $\varphi \in \mathcal{S}_0'$ by the condition $\displaystyle\int_{\mathbb R^n}\varphi \, d\boldsymbol{x}=1$. An equation is said to satisfy the sum rules of order $\ell \geqslant 0$ if the $m-1$ points $\boldsymbol{\xi}=M^{-\top}\boldsymbol{d}_k^*$, $\boldsymbol{d}_k^* \in D^*\setminus \{\boldsymbol{0}\}$, are zeros of order $\geqslant \ell+ 1$ of the mask $\boldsymbol{a}(\boldsymbol{\xi})$, where $D^*$ is a set of digits for the matrix $M^\top$. The set $D^*$ can be different from $D$. A refinable function $\varphi$ is stable if its integer shifts $\{\varphi(\cdot-\boldsymbol{k})\}_{\boldsymbol{k} \in \mathbb Z^n}$ are linearly independent. We make these standard assumptions throughout. Any tile $\mathrm B$-spline $B_{\ell}^G$ of order $\ell$ is stable and satisfies the sum rules of order $\ell$ [7]. In the one-dimensional case the smoothness of refinable functions has extensively been studied and two methods have been used widely. In the matrix approach (see [1], [5], [15] and [42]) one finds the largest $k$ such that $\varphi\in C^k(\mathbb R)$, and then looks for the sharp value of the Hölder exponent in this space. Definition 5. The general Hölder smoothness exponent of a function $\varphi$ in the space $C$ is defined by where $k$ is the largest integer such that $\varphi \in C^k(\mathbb R^d)$. For $\varphi \in C^{\infty}$ we set $\alpha_{\varphi}=+\infty$. Remark 1. Similarly to the smoothness exponent in $C$, the one in $L_2$ is defined by replacing $C^k(\mathbb R^d)$ by $W_2^k(\mathbb R^d)$. In the matrix method the joint spectral radius of special matrices must be calculated, which is an involved problem even for small dimensions [2], [12], [13]. This method can also be used to find smoothness in $L_2(\mathbb R)$ [40] and in the Sobolev space $W_2^k(\mathbb R)$ (of absolutely continuous functions $f$ with $f^{(k)} \in L_2$). In this case the calculation of the joint spectral radius is reduced to finding the usual spectral radius but for a larger matrix of size of about $\frac12N^2$ [4]. In another approach, which is usually called the calculation of Sobolev smoothness or the Littlewood–Paley method, we find the Sobolev smoothness defined by
$$
\begin{equation*}
\sup\biggl\{ \beta \geqslant 0 \colon\int_{\mathbb R}|\widehat \varphi(\xi)|^2 (1+ |\xi|^2)^{\beta}\, dt<\infty\biggr\}.
\end{equation*}
\notag
$$
It is known that in $L_2$ the Sobolev smoothness exponent coincides with the Hölder one (see Remark 1). In particular, if the Sobolev smoothness exceeds $k$, then $\varphi \in W_2^k(\mathbb R)$. This problem is reduced to calculating the usual spectral radius of a matrix of much smaller size, of about $2N$ (see [25], [26] and [47]). The main disadvantage of this method is that it calculates smoothness in $L_2$, rather than in $C$. Extensions of both methods to the multivariate case have been known for a long time (see [14], [15], [21], [29], [30], [32], [34], [35], [38] and [45]) but only for isotropic matrices. There are simple examples showing that these extensions do not work in the nonisotropic case. Nonisotropic matrices appear in applications (see [9], [10], [17]–[20] and [38]). In [10], [16] and [22] the smoothness was calculated in some special anisotropic spaces, and in [33] Sobolev smoothness was estimated. In 2019 a modification of the matrix approach was proposed for nonisotropic dilations [20]. However, because of the large size of matrices, this method is mainly used for quite small dimensions and a small number of coefficients. Recently the Littlewood–Paley method was extended to the case of nonisotropic matrices [50]. Below we present this method briefly and consider in detail its modification for tile $\mathrm B$-splines. Namely, we discuss the geometrical conditions under which we can write a simple formula similar to the isotropic case. Using this approach we can evaluate the smoothness of tile $\mathrm B$-splines for large dimensions and produce many new supersmooth tile $\mathrm B$-splines. New numerical results are discussed in § 6. Assume that we need to evaluate the smoothness of a refinable function $\varphi$. Let $W$ be the linear span of the set $\{\varphi(\cdot +\boldsymbol{h})-\varphi (\,{\cdot}\,) \mid \boldsymbol{h} \in \mathbb R^n\}$. In the Littlewood—Paley method, with each function $f\in W$ we associate the periodization
$$
\begin{equation}
\Phi_f(\boldsymbol{\xi})=\sum_{\boldsymbol{k} \in \mathbb Z^n} \bigl| \widehat{f}(\boldsymbol{\xi}+\boldsymbol{k}) \bigr|^2
\end{equation}
\tag{3.1}
$$
of its Fourier transform $\widehat f$. The Plancherel formula implies that $\|\Phi_f\|_1\,{=}\,\|\widehat f\|_2^2\!=\!\|f\|_2^2$ (the norm of $\Phi_f$ is taken in $L_1({\mathbb T}_n)$). For any function $f\in W$ with compact support the function $\Phi_{f}$ is a trigonometric polynomial. Indeed, $\Phi_{f}$ expands in a Fourier series as follows:
$$
\begin{equation*}
\Phi_{f}(\boldsymbol{\xi})=\sum_{\boldsymbol{k} \in \mathbb Z^n}\bigl(f(\,{\cdot}\,), f(\cdot+\boldsymbol{k})\bigr)e^{2\pi i(\boldsymbol{k}, \boldsymbol{\xi})},
\end{equation*}
\notag
$$
where only a finite number of coefficients $\bigl(f(\,{\cdot}\,), f(\cdot+ \boldsymbol{k}) \bigr)$ can be distinct from zero. Consider the symmetrized tile $\mathrm B$-spline $\varphi * (\varphi\_)$, where $\varphi\_(\,{\cdot}\,)=\varphi (-\,\,{\cdot}\,)$. This $\mathrm B$-spline has a centrally symmetric support, whose intersection with the set of integer points is denoted by $\Omega$. For each $f \in W$ the spectrum of the trigonometric polynomial $\Phi_f$ is a subset of $\Omega$. We define the operator $\mathcal{T}$ on $L_2({\mathbb T}_n)$ as follows. Let $D^*= \{\boldsymbol{d}_i^*\}_{i=0}^{m-1}$ be a set of digits for the transposed matrix $M^\top$. Then
$$
\begin{equation*}
\bigl[\mathcal{T} \boldsymbol{p} \bigr](\boldsymbol{\xi})= \sum_{j=0}^{m-1}\bigl|\boldsymbol{a}(M^{-\top}(\boldsymbol{\xi}+\boldsymbol{d}^*_j)) \bigr|^2 \boldsymbol{p} \bigl(M^{-\top}(\boldsymbol{\xi}+\boldsymbol{d}^*_j)\bigr), \qquad \boldsymbol{p} \in L_2({\mathbb T}_n) .
\end{equation*}
\notag
$$
Example 3. If $M=2$ and $D^*=\{0, 1\}$, then The above operator $\mathcal{T}$ maps the polynomial space $\mathcal{P}_{\Omega}$ into itself (see [1], [26] and [32]), and in what follows we restrict $\mathcal{T}$ to this subspace. We denote the matrix of the operator on this subspace by $T$. Calculating it, we arrive at the following matrix. For all $\omega_i,\omega_j \in \Omega$, if the shift by $M \omega_i-\omega_j$ enters the refinement equation for $\varphi * (\varphi\_)$ with nonzero coefficient $c$, then $T_{i, j}=c$ (see [1] for one variable; the multivariate case is dealt with similarly). Example 4. Consider now the following two-dimensional example: The tile $G$ is a direct product of one-dimensional closed intervals (that is, a unit square). Now, for a spline of order 0 we can find the set such that the operator $\mathcal{T}$ maps the space of polynomials $\mathcal{P}_{\Omega}$ into itself. The operator $\mathcal{T}$ can be applied to $\Phi_f$ several times. The asymptotic decay of the sequence $\mathcal{T}^k[\Phi_f]$ as $k \to +\infty$ controls the smoothness of the function $\varphi$. Namely, if $\|\mathcal{T}^k[\Phi_f]\|_1 \asymp \lambda^k$, then the Hölder exponent in $L_2$ is $\alpha_{\varphi} \leqslant \log_{1/r} \lambda$ (see, for example, [50]). For one variable this result was proved in [26]. We denote by $\mathcal{P}_0=\mathcal{P}_0(\Omega)$ the space of polynomials with zero of multiplicity ${\geqslant 2}$ at $\boldsymbol{\xi}=0$. For the refinement equations with isotropic matrix $M$, the Hölder exponent of $\varphi$ in $L_2$ is $ \log_{1/r} \rho(\mathcal{T}|_{\mathcal{P}_0})$ (see, for example, [30]). The proof is based on the fact that in the isotropic case, if $f\in W$, then $\Phi_f \in \mathcal{P}_0$. For a general matrix $M$ without isotropy condition this is not true in general, and the formula for smoothness in $L_2$ fails (the corresponding examples are well known; see [20]). This case was studied in [50], where a formula for smoothness in $W_2^k(\mathbb R^n)$ was obtained for an arbitrary refinement equation with integer expanding matrix $M$. In this paper we apply this formula to tile $\mathrm B$-splines. Then we show that, under fairly general assumptions, the ‘simple formula’ $ \log_{1/r} \rho(\mathcal{T}|_{\mathcal{P}_0})$ holds for the smoothness exponent in $L_2$, and there is also a similar higher-order result on smoothnesses (Theorem 2). We also state sufficient conditions, in geometric terms and in terms of the matrix $M$, for the ‘simple case’ to hold. Given $\ell \geqslant 0$, we let $\mathcal{P}_\ell$ denote the set of polynomials $\boldsymbol{p} \in \mathcal{P}$ with zero of order $\geqslant 2(\ell+1)$ at $\boldsymbol{\xi}=0$, that is, $\boldsymbol{p}(\boldsymbol{0})=\boldsymbol{p}'(\boldsymbol{0})=\cdots= \boldsymbol{p}^{(2\ell+1)}(\boldsymbol{0})=0$. (The equality $\boldsymbol{p}^{(k)}(\boldsymbol{x})=0$ means that all partial derivatives of order $k$ vanish at the point $\boldsymbol{x}$.) The mask $\boldsymbol{a}$ of the tile $\mathrm B$-spline $B_{\ell}^G$ satisfies the sum rules of order $\ell$, and so each subspace $\mathcal{P}_{k}$, $k=0, 1, \dots, \ell$, is also $\mathcal{T}$-invariant (see, for example, [30] and [35]). We denote the spectral radius of the corresponding restriction by $\rho_k= \rho(\mathcal{T}|_{\mathcal{P}_k})$. Let $J_1$ be the spectral subspace of the matrix $M$ corresponding to its largest (in modulus) eigenvalue $r=r_1$. Definition 6. Let $J \subset \mathbb R^n$ be a fixed subspace. We call $\boldsymbol{\xi}$ a periodic point of a function $f$ if for each $\boldsymbol{k} \in \mathbb Z^n$ we have $f(\boldsymbol{\xi}+\boldsymbol{k})=0$ or $\boldsymbol{\xi}+ \boldsymbol{k} \in J^{\perp}$. Note that if $\varphi$ is a refinable function and $J=J_1$, then $\widehat \varphi$ always has the periodic point $\boldsymbol{0}$. Indeed, for $\boldsymbol{k}=\boldsymbol{0}$, we have $\boldsymbol{k} \in J_1^{\perp}$, and for $\boldsymbol{k} \ne \boldsymbol{0}$ we have $\widehat \varphi(\boldsymbol{k})=0$ (this follows from the sum rules [15]). Hence any integer point is periodic. The following theorem (see [50]) establishes a link between the smoothness of the refinable function and periodic points. In what follows we assume that $J=J_1$. Theorem A. If the Fourier transform of a tile $\mathrm B$-spline $\widehat B_\ell^G$ has no noninteger periodic points, then its smoothness satisfies where $r=\rho(M)$ is the spectral radius of $M$ and $\rho_\ell= \rho(\mathcal{T}|_{\mathcal{P}_\ell})$. Formula (*) will be referred to as the ‘simple formula’, because it is indeed a simple particular case of the general formula for smoothness in [50]. Thus, we have the following algorithm for the evaluation of smoothness in the ‘simple case’:
Now we go over to some auxiliary objects, which will be used for the geometric interpretation of the assumptions of Theorem A. § 4. Tile cylinders and rational spacesBy a cylinder $C$ parallel to a subspace $J \subset \mathbb R^n$ we mean a nonempty closed subset of $\mathbb R^n$ which is invariant under all translations parallel to $J$. In particular, if $J=\mathbb R^n$, then $C=\mathbb R^n$; if $J=\{\boldsymbol{0}\}$, then $C$ is an arbitrary closed set. However, we will not consider such cases and always assume that $J$ is a nontrivial proper subspace of $\mathbb R^n$. We say that a tiling $\{G+\boldsymbol{k}\}_{\boldsymbol{k} \in \mathbb Z^n}$ contains a cylinder $C$ if there exists a subset $E $ of $\mathbb Z^n$ such that $C=\bigcup_{\boldsymbol{k} \in E} (G+\boldsymbol{k})$. A cylinder contained in a given tiling is called a tile cylinder. The set of all tile cylinders parallel to a subspace $J$ is closed under the operations of intersection, union and difference (for the last two operations, one must take the closures first). Each tiling contains at least one cylinder parallel to $J$; this is the cylinder $C=\mathbb R^n$. This case is trivial. Other tile cylinders (if exist) are called proper cylinders. The tiling from Example 2 (see Figure 1) contains a (horizontal) proper cylinder. A tile cylinder is minimal if it contains no other tile cylinders. We can also fix a subspace $J$ and consider a minimal tile cylinder among the ones parallel to $J$. Lemma 1. Given an arbitrary tiling and a subspace $J\subset \mathbb R^n$, there exists a minimal tile cylinder parallel to $J$. Proof. We choose a tile $G$ and consider the intersection $C$ of all cylinders parallel to $J$ and containing $G$. This cylinder is not minimal if it contains another cylinder $C'$; then $C'$ does not contain $G$. However, in this case the closure of the set $C\setminus C'$ is a cylinder containing $G$, which contradicts the definition of $C$. Therefore, $C$ is minimal.
This proves the lemma. Lemma 2. All minimal tile cylinders parallel to $J$ are integer translates of one another. Any tile cylinder parallel to $J$ can be partitioned into minimal ones. Proof. The intersection of all tile cylinders parallel to $J$ and containing the tile $G$ is a minimal cylinder (see the proof of Lemma 1), which we denote by $C$. Any other minimal cylinder $C'$ contains some tile $G'=G + \boldsymbol{k}$. Therefore, the cylinder $C' - \boldsymbol{k}$ contains $G$, and therefore it contains $C$ because $C$ is minimal. If $C$ is a proper subset of $C' - \boldsymbol{k}$, then the cylinder $C + \boldsymbol{k}$, which contains $G + \boldsymbol{k}=G'$, is also a proper subset of $C'$. However, this is impossible because $C'$ is minimal.
Thus, $C=C'-\boldsymbol{k}$, and we have proved that all minimal cylinders are obtained from each other by integer translation. Now we can introduce an equivalence relation on the set of tiles: two tiles are equivalent if they lie in the same minimal cylinder. If $C_0$ is an arbitrary tile cylinder parallel to $J$, then we take in it a maximal system of inequivalent tiles. Then the corresponding minimal cylinders cover $C_0$. This proves the lemma. The space $\mathbb R^n$ is also a cylinder, and so we have the following result. A linear subspace $L \subset \mathbb R^n$ of dimension $d$ is called a rational space if it is a span of some integer vectors. It is known that each rational subspace has a basis of integer vectors such that the integer linear combinations of elements of this basis contain the whole integer part of $L$, that is, the lattice $L\cap \mathbb Z^n$. Such a basis will be called an elementary basis. The characteristic property of an elementary basis is that the determinant of its Gram matrix is $1$. Indeed, this condition means that the $d$-dimensional volume of the parallelepiped spanned by the vectors in the basis is 1, and so this parallelepiped generates the integer lattice of $L$. Each elementary basis can be augmented to an elementary basis of $\mathbb R^n$ which generates the lattice $\mathbb Z^n$. There exists an integer matrix that takes the elementary basis to the first $d$ vectors of the standard basis of $\mathbb R^n$. Both the linear span and the intersection of several rational subspaces is a rational subspace. Hence for each subspace $J $ of $\mathbb R^n$ there exists an inclusion-minimal rational subspace $L$ containing $J$. This subspace is called a minimal rational subspace containing $J$. The following result holds [50]. Proposition 1. Any minimal tile cylinder parallel to a subspace $J$ is parallel to some minimal rational subspace containing $J$. In view of Proposition 1 we can consider only rational subspaces parallel to cylinders. The linear hull of several subspaces parallel to a cylinder is also parallel to it, and so there exists an inclusion-maximal subspace parallel to this cylinder; this subspace will be called the generator. From Proposition 1 it follows that the generator of a minimal tile cylinder $C$ contains all (not necessarily rational) subspaces parallel to $C$. Corollary 2. The generator of a minimal tile cylinder parallel to a subspace $J$ is a rational subspace. § 5. Smoothness of tile $\mathrm B$-splinesIn this section we employ the auxiliary result from § 4 to derive a key smoothness theorem for tile $\mathrm B$-splines. Theorem 1. Let $\varphi=B_\ell^G$ be a tile $\mathrm B$-spline of order $\ell$ generated by a tile $G$. Assume that the tiling $\{G+\boldsymbol{k}\}_{\boldsymbol{k} \in \mathbb Z^n}$ does not contain a proper cylinder parallel to a minimal rational subspace containing $J_1$. Then the Fourier transform of the function $\varphi(\boldsymbol{x})$ has no noninteger periodic points. Proof. Assume for a contradiction that there exists a noninteger periodic point $\mathbf z$ of the Fourier transform $\widehat \varphi$. By stability $\widehat \varphi$ cannot have a periodic zero (see [15]), so translating the point $\mathbf z $ by an integer vector if necessary we can assume that $\mathbf z \in J_1^{\perp}$. By the definition of a periodic point, for each integer $\boldsymbol{k}$ either the vector $\mathbf z+\boldsymbol{k}$ is orthogonal to $J_1$, or $\widehat \varphi(\mathbf z+ \boldsymbol{k})=0$.
Consider the indicator of the tile $\phi=\chi_{G}$. Since $\widehat \varphi=(\widehat{\phi})^{\ell}$, the functions $\widehat \varphi$ and $\widehat \phi$ have the same zeros, and now the last condition can be written as $\widehat \phi(\mathbf z+ \boldsymbol{k})=0$. For each $\boldsymbol{h} \in J_1$ consider the generalized derivative $\phi_{\boldsymbol{h}} \in \mathcal{S}'$ (the generalized derivative along the vector $\boldsymbol{h}$). For each integer $\boldsymbol{k} \in \mathbb Z^n$ either $(\boldsymbol{h}, \mathbf z+\boldsymbol{k})=0$, or $\widehat \phi(\mathbf z+\boldsymbol{k})=0$. As a result, we have
$$
\begin{equation*}
\widehat{\phi_{\boldsymbol{h}}}(\mathbf z+\boldsymbol{k})=2\pi i(\boldsymbol{h}, \mathbf z+\boldsymbol{k})\widehat{\phi}(\mathbf z+\boldsymbol{k}) =0 \quad \text{for all } \boldsymbol{k} \in \mathbb Z^n.
\end{equation*}
\notag
$$
Let us use Poisson’s summation formula. For each $\boldsymbol{x} \in \mathbb R^n$,
$$
\begin{equation*}
\sum _{\boldsymbol{q} \in \mathbb Z^n} f(\boldsymbol{x}+\boldsymbol{q})=\sum _{\boldsymbol{k} \in \mathbb Z^n} \widehat f(\boldsymbol{k}) e^{2\pi i (\boldsymbol{k}, \boldsymbol{x})},
\end{equation*}
\notag
$$
where we plug in $f(\boldsymbol{x})=\phi_{\boldsymbol{h}} (\boldsymbol{x}) e^{-2\pi i (\mathbf z, \boldsymbol{x})}$. Hence $\widehat{f}(\boldsymbol{x})= \widehat{\phi_{\boldsymbol{h}}}(\mathbf z+\boldsymbol{x})$ and, consequently,
$$
\begin{equation*}
\sum _{\boldsymbol{q} \in \mathbb Z^n} \phi_{\boldsymbol{h}} (\boldsymbol{x}+ \boldsymbol{q}) e^{-2\pi i (\mathbf z, \boldsymbol{x}+\boldsymbol{q})}=\sum _{\boldsymbol{k} \in \mathbb Z^n} \widehat{\phi_{\boldsymbol{h}}}(\mathbf z+\boldsymbol{k}) e^{2\pi i (\boldsymbol{k}, \boldsymbol{x})}=0.
\end{equation*}
\notag
$$
Therefore, $\sum_{\boldsymbol{q} \in \mathbb Z^n} e^{2\pi i (\mathbf z, \boldsymbol{q}) }\phi_{\boldsymbol{h}}(\cdot+\boldsymbol{q})= 0$. Consider the function
$$
\begin{equation}
F(\boldsymbol{x}) =\sum_{\boldsymbol{q} \in \mathbb Z^n} e^{-2\pi i (\mathbf z, \boldsymbol{q})}\phi(\boldsymbol{x}+\boldsymbol{q}).
\end{equation}
\tag{5.1}
$$
This sum is well defined because $\phi$ has support on a compact set (a tile). Indeed, $F$ is a piecewise function which is constant on elements of the tiling $\{G + \boldsymbol{k}\}_{\boldsymbol{k} \in \mathbb Z^n}$. The set of coefficients $\{e^{-2\pi i ( \mathbf z, \boldsymbol{q})}\}_{\boldsymbol{q} \in \mathbb Z^n}$ is bounded, and so the series also converges in $\mathcal{S}'$. Hence this series can be differentiated termwise along $\boldsymbol{h}$. Therefore, $F_{\boldsymbol{h}}=0$, and $F$ is constant on any straight line parallel to $\boldsymbol{h}$. The vector $\boldsymbol{h}\in J_1$ is arbitrary, so $F$ is also constant on any affine plane with linear part $J_1$. On the other hand $F$ is piecewise constant on the tiling, and so any level set of $F$ is either empty or is a cylinder parallel to $J_1$. The vector $\mathbf z$ is noninteger, and so not all coefficients $e^{-2\pi i ( \mathbf z, \boldsymbol{q})}$, $\boldsymbol{q}\in \mathbb Z^n$, are equal. Hence level sets of $F$ are proper cylinders. Now an application of Proposition 1 completes the proof of Theorem 1. Recall that $\mathcal{P}_\ell$ is the set of polynomials $\boldsymbol{p} \in \mathcal{P}$ with zero of order $\geqslant 2(\ell+1)$ at ${\boldsymbol{\xi}=0}$. Let us now formulate the resulting smoothness theorem for tile $\mathrm B$-splines. Theorem 2. Given a tile $\mathrm B$-spline $B_\ell^G$, if the corresponding tiling $\{G + \boldsymbol{k}\}_{\boldsymbol{k} \in \mathbb Z^n}$ contains no proper cylinder parallel to a minimal rational subspace $L_1$ containing the space $J_1$, then the smoothness exponent satisfies the ‘simple formula’ where $r=\rho(M)$ is the spectral radius of $M$ and $\rho_\ell=\rho(\mathcal{T}|_{\mathcal{P}_\ell})$. Each of the following two conditions implies the ‘simple formula’: (1) $L_1$ coincides with $\mathbb R^n$; (2) there is no integer basis of $\mathbb R^n$ in which the matrix $M$ is represented as a block upper triangular integer matrix with first block $A$ corresponding to $L_1$: Proof. By Theorem 1, if there is no proper cylinder, then the Fourier transform of a tile $\mathrm B$-spline has no periodic points, and so by Theorem A we have the ‘simple formula’.
If the tiling $\{G+\boldsymbol{k}\}_{\boldsymbol{k} \in \mathbb Z^n}$ contained a proper cylinder parallel to $L_1$, then $L_1$ would be different from $\mathbb R^n$, which proves (1). Let us verify (2). Assume for a contradiction that there exists a proper cylinder parallel to $L_1$. Since $L_1$ is a minimal rational space containing $J_1$, we have $J_1 \subset L_1$. By the definition of $J_1$, $MJ_1= J_1$. Hence $ML_1$ is also a rational space for $J_1$. As a result, $ML_1$ contains a minimal space $L_1$. So $L_1 \subset ML_1$, and therefore $L_1=ML_1$, because the matrix $M$ is nonsingular. We choose an elementary basis of $L_1$ and augment it to an elementary basis of $\mathbb R^n$. This gives us a basis in which $M$ is an integer matrix of the form (5.2), which is a contradiction. Theorem 2 is proved. The following result was proved in [50]. Theorem B. For a bivariate tile $\mathrm B$-spline $B_\ell^G$ the inequality can only hold for $m \geqslant 6$. § 6. SupersmoothnessRecall that a tile $\mathrm B$-splines is supersmooth if its smoothness exceeds that of classical $\mathrm B$-splines of the same order, that is, $\alpha (B_{\ell}^G)>\ell+1/2$. Any tile generating such a $\mathrm B$-spline will also be referred to as a supersmooth tile (even though the smoothness of a such a tile can be low). That supersmooth $\mathrm B$-splines exist is quite unexpected, because the smoothness of the characteristic functions of their generating tile is always lower than that of the characteristic function of the cube, which generates classical $\mathrm B$-splines. Nevertheless, after several convolutions tile $\mathrm B$-splines can ‘outrun’ classical ones. Numerical results show that supersmoothness occurs quite rarely. Our conjecture is that there are 20 supersmooth bivariate tiles with number of digits at most five (here we identify tiles generating splines of equal smoothness). These tiles are listed in Table 1. Table 1.The smoothness exponents of the supersmooth bivariate tile $\mathrm B$-splines known to the author
In this section, a ‘tile’ is understood in a slightly more general sense, namely, we assume that its Lebesgue measure can be an arbitrary integer (this has no effect on the evaluation of smoothness). First we consider the case of two-digit tiles, so that $m = |{\det M}| = 2$. The smaller the number of digits, the fewer coefficients the refinement equation has, and so the smaller computational costs of the applied algorithms (namely, the subdivision and cascade ones). It is known that there exist three two-digit tiles in the plane (up to affine similarity) [11]: ‘Bear’ (tame twindragon), ‘Dragon’ (twindragon), and ‘Square’. A change of digits in a two-digit tile transforms it into an affinely similar tile [8], which is not the case for tiles with a larger number of digits. The smoothness of tile $\mathrm B$-splines constructed from plane two-digit tiles was examined in [7] and [50]. The ‘Bear’ tile (the second line of Table 1) generates supersmooth splines, in contrast to the ‘Dragon’ tile. Any spline generated by the ‘Dragon’ tile has smoothness lower than classical splines (the corresponding results are known up to order $9$). The ‘Square’ tile generates splines equal to standard ones. In Figure 2 we show a partition of the ‘Bear’ tile for the matrix $M=((1, -2), (1, 0))$ and digits $D=\{(0, 0), (1, 0)\}$ into two parts similar to this tile. Below we obtain some auxiliary theoretical results on the smoothness of tile $\mathrm B$-splines and tiles. These results will be applied to the search of supersmooth splines. Two matrices $M_1$ and $M_2$ are called $\mathbb Z$-similar if there exists an integer matrix $C$, $|{\det{C}}|=1$, such that $M_1=C^{-1} M_2 C$. This is an equivalence relation on the set of integer matrices. Lemma 3. Let two expanding integer matrices $M_1$ and $ M_2$ be $\mathbb Z$-similar. Then for each tile $G_1$ generated by $M_1$ there exist an affinely similar tile $G_2$ generated by $M_2$. Proof. By $\mathbb Z$-similarity, $M_1=C^{-1} M_2 C$ for some matrix $C \in {\mathbb Z}^{n \times n}$ such that $C^{-1} \in {\mathbb Z}^{n \times n}$. Let the tile $G_1=G_1(M_1, D_1)$ be generated by a set of digits $D_1$. Then
$$
\begin{equation*}
\begin{aligned} \, G_1&=\biggl\{ \sum _{k=1}^{\infty}M_1^{-k}\boldsymbol{\Delta}^1_k \biggm| \boldsymbol{\Delta}^1_k \in D_1\biggr\} =\biggl\{ \sum _{k=1}^{\infty}C^{-1} M_2^{-k} C \boldsymbol{\Delta}^1_k \biggm| \boldsymbol{\Delta}^1_k \in D_1\biggr\} \\ &=C^{-1} \biggl\{ \sum _{k=1}^{\infty} M_2^{-k} \boldsymbol{\Delta}^2_k \biggm| \boldsymbol{\Delta}^2_k \in D_2\biggr\}=C^{-1} G_2, \end{aligned}
\end{equation*}
\notag
$$
where $D_2=C D_1$, and $G_2=G_2(M_2, D_2)$ is the tile generated by $M_2$. It only remains to see that the choice of digits $D_2 \subset\mathbb Z^n$ is admissible. If $C\boldsymbol{d}_1-C\boldsymbol{d}_2=M_2 \mathbf z$ for some integer $\mathbf z$, then $\boldsymbol{d}_1-\boldsymbol{d}_2= C^{-1} C M_1 C^{-1} \mathbf z=M_1 (C^{-1} \mathbf z)$. Since $D_1$ is a set of digits and $C^{-1}\mathbf z \subset \mathbb Z^n$, this is possible only for $\boldsymbol{d}_1=\boldsymbol{d}_2$, so that $C\boldsymbol{d}_1= C\boldsymbol{d}_2$. This shows that the tiles $G_1$ and $G_2$ are affinely similar.
Lemma 3 is proved. For two-digit tiles a change of digits produces an affinely similar tile, and therefore the smoothness of such tile $\mathrm B$-splines depends only on the matrix. If a tile has more than two digits, then its smoothness can vary with a change of digits. Given a matrix $M$, we consider all possible digits and the corresponding tiles and construct all possible tile $\mathrm B$-splines of fixed order $\ell$. Such splines will be referred to as families of tile $\mathrm B$-splines of order $\ell$ corresponding to the matrix $M$. We are interested in the range of their smoothness exponents — the set of smoothness exponents ‘attained’ at splines of prescribed order with the given matrix $M$. Once we know this range, we can check if supersmooth tiles exist for a given matrix. We show below that for $\mathbb Z$-similar matrices, the corresponding families of tile $\mathrm B$-splines have the same range of smoothness exponents, and then we use the results from [36] on classes of $\mathbb Z$-similar matrices with few digits ($m=3, 4, 5$) on the plane. So, instead of going through all possible expanding matrices with a fixed number of digits (the determinant of the matrix), it is sufficient to consider only one representative of each class. In this way we will find apparently all supersmooth tiles with few digits (our exhaustive search is not complete because of infinitely many possible sets of digits, see below). Corollary 3. Given $\mathbb Z$-similar matrices, the corresponding families of tile $\mathrm B$-splines have equal ranges of smoothness exponents. Proof. Fix two $\mathbb Z$-similar matrices $M_1$ and $M_2$. Consider the tile $\mathrm B$-spline $B_\ell^{G_1}$, where the tile $G_1=G_1(M_1, D_1)$ is taken from the first family. By Lemma 3 there exists an affinely similar tile $G_2=G_2(M_2, D_2)$. The splines $B_\ell^{G_1}$ and $B_\ell^{G_2}$ constructed from the affinely similar tiles $G_1$ and $G_2$ have equal smoothness, which proves the claim. Each expanding integer polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ corresponds to at least one expanding matrix, namely, its companion matrix
$$
\begin{equation}
M=\begin{pmatrix} 0 & 0 & \cdots & 0 &-a_0\\ 1 & 0 & \cdots & 0 & -a_{1} \\ 0 & 1 & \cdots & 0 & -a_{2} \\ \vdots & \vdots & \ddots & \ddots& \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1}, \end{pmatrix},
\end{equation}
\tag{6.1}
$$
that is, $p$ is the characteristic polynomial of the matrix $M$. Now consider bivariate tile $\mathrm B$-splines. In the bivariate case, for $m \leqslant 5$ we can use the ‘simple formula’ from Theorem B, and so we consider only this case. The two-digit case ($m=2$) was treated in [7]; in this case, as already mentioned, of the three tiles ‘Bear’, ‘Square’ and “Dragon’, only the ‘Bear’ tile is supersmooth. So, in what follows we consider $m=3, 4, 5$. Matrices with different spectra are not similar, and so at least the companion matrices of all possible expanding polynomials of the form $x^2 + ax\pm m$ for some $a \in \mathbb Z$ (polynomials with roots outside the unit disc) lie in different classes of $\mathbb Z$-similarity. A description of such polynomials is quite simple (see [11]); however, we provide another argument. Lemma 4. Let $m \in \mathbb N, m > 1$, and $a \in \mathbb Z$. Then: (1) the polynomial $x^2+ax+m$ is expanding if and only if $|a| \leqslant m$; (2) the polynomial $x^2+ax-m$ is expanding if and only if $|a| \leqslant m- 2$. Proof. We can employ the Schur–Cohn algorithm. For a real polynomial $p(x)=a_nx^n+\dots+a_0$ consider the polynomial $p^*(x)=a_0 x^n+\dots+a_n$; the Schur transformation is defined as the polynomial $Tp=p(0) p-p^*(0) p^*$, which has degree one less than $p^*(x)$.
In case (1), andIn case (2), andSchur [46] proved that if $|a_0| > |a_n|$ (this inequality holds in both cases because $m > 1$), then the polynomials $p(x)$ and $Tp(x)$ are either both expanding or both not expanding. So it suffices to check when $Tp$ is expanding. In case (1) the modulus of the unique root $-(m+1) / a$ exceeds $1$ if and only if $|a| < m+1$, that is, $|a| \leqslant m$. In case (2) the modulus of the unique root $-(m-1) / a$ is greater than $1$ if and only if $|a| < m-1$, that is, $|a| \leqslant m-2$. This proves the lemma. However, not all expanding matrices of size $2 \times 2$ with determinants $\pm 3, \pm 4, \pm 5$ are $\mathbb Z$-similar to one of such companion matrices. A description of all classes of $\mathbb Z$-similarity for $m=3, 4, 5$ can be found in Table 3 in [36]; this description was obtained by classifying ideals of $\mathbb Z[\theta]$. In the case $m=3$ one class of matrices which are $\mathbb Z$-similar to the matrix $((1, 2), (-2, -1))$ is added. Thus, to find supersmooth tiles it suffices to study the tile $\mathrm B$-splines generated by the companion matrices of the polynomials $x^2-ax+m$, $0 \leqslant a \leqslant m$, and $x^2-ax-m$, $0 \leqslant a \leqslant m-2$ ($m=3, 4, 5$), and also by the special matrices from Table 3 in [36]. The number of polynomials can be reduced even further using symmetry. We give an argument valid for an arbitrary dimension. Definition 7. Algebraic polynomials $p=\sum _{k=0}^{n} p_k t^k$ and $q=\sum _{k=0}^{n} q_k t^k$ are called opposite if $q_k=(-1)^{n-k}p_k$, $k=0, \dots, n$. The leading coefficients of two opposite polynomials are equal. By Viète’s theorem the zeros of the polynomial $q$ are opposite to those of $p$. The matrices $M$ and $-M$ have opposite characteristic polynomials. Remark 2. For two-digit matrices $M$ opposite polynomials generate the same tile (for an appropriate choice of digits) [8]. This property is based on the fact that each two-digit tile is centrally symmetric. However, this is not so in the general case, and the tiles constructed from opposite polynomials are not necessarily affinely similar. Nevertheless, we show below that they have equal smoothness, which is also supported by numerical experiments. Theorem 3. Given two matrices with opposite polynomials, the corresponding families of $\mathrm B$-splines have the same smoothness exponents. Proof. Consider two tiles $G_1=G(M, D)$ and $G_2=G(-M, D)$. We claim that the tile $\mathrm B$-splines constructed from these tiles have the same smoothness.
We follow the algorithm in § 3 for smoothness evaluation step by step. These tiles have equal masks in the refinement equation for the symmetrized $\mathrm B$-spline, because both the digits and determinant of the matrix are equal. Since $\Omega$ is centrally symmetric, it follows from the algorithm of its construction that these two splines have equal sets $\Omega$. For all $\omega_i, \omega_j \in \Omega$, if the shift by $M \omega_i-\omega_j$ enters the refinement equation for $\varphi * (\varphi\_)$ with nonzero coefficient $c$, then $T_{i, j}=c$. In the case of the second matrix $-M$ the same coefficient corresponds to the pair $-\omega_i,\omega_j \in \Omega$, because $M \omega_i-\omega_j=(-M) (-\omega_i)-\omega_j$ since $\Omega$ is centrally symmetric. Thus, all differences between the two cases consist in permutations of elements of $\Omega$ and the matrix $T$, which has no effect on smoothness. This proves the theorem. Let us indicate the results of a numerical examination of cases. For each $m=3, 4, 5$ and each polynomial $x^2 - ax + m$, $0 \leqslant a \leqslant m$, or $x^2 - ax - m$, $0 \leqslant a \leqslant m - 2$, consider its companion matrix. We also consider the matrices from [36]. For each possible digit we look at a few possible variants, consider all possible combinations of these variants and evaluate the resulting smoothness exponents of tile $\mathrm B$-splines of lower order. The number of digits is infinite, and so we cannot be sure that we perform an exhaustive search; however, numerical experiments suggest that there is no supersmoothness for large digits. Similarly, for $m= 4$ the number of classes for $(x \pm 2)^2$ is infinite, but the highest smoothness comes from parallelograms. In the same way we can limit ourselves to splines of low order. This takes us to the following conjecture. Conjecture 1. All possible smoothness exponents of supersmooth tile $\mathrm B$-splines with number of digits $\leqslant 5$ are shown in Table 1. The first line of Table 1 shows, for comparison, the values of smoothness exponents of the classical $\mathrm B$-splines. For five-digit tiles we do not indicate their common digit $(0, 0)$. If the characteristic polynomial is indicated, then the corresponding matrix is its companion matrix. Among these supersmooth splines, there are no splines generated by nonisotropic matrices — numerical results show that the smoothness of these splines is much lower. Conjecture 2. There are no nonisotropic supersmooth tile $\mathrm B$-splines with $\leqslant 5$ digits. Two three-digit supersmooth splines in the above list are the ‘ThreeDig1’ and ‘ThreeDig2’ splines from [50], but with different matrices and digits:
$$
\begin{equation*}
M_{1}=\begin{pmatrix}1 & -2 \\ 1 & 1\end{pmatrix}, \qquad M_{2}=\begin{pmatrix}1 & -1 \\ 1 & 2\end{pmatrix}\quad\text{and} \quad D=\bigl\{ (0,0), (1,0), (0,1)\bigr\}.
\end{equation*}
\notag
$$
These splines correspond to the polynomials $x^2-2x+3$ and $x^2-3x+3$, respectively. Remark 3. There is a matrix with different spectrum which gives the same values of smoothness exponents as the supersmooth tile with polynomial $x^2-3x+3$, even though the corresponding tiles are not affinely similar. This is the companion matrix for $x^2-3$ with digits $\{(0, 0), (1, 0), (-1, 1)\}$. Now we mention some properties of supersmooth tiles. It is known that a stable refinable function $\varphi_{\ell}$ satisfying the sum rules of order $\ell$ cannot lie in $W^{\ell+1}_2$, and so $\alpha (\varphi_{\ell}) < \ell+1$. We can see that for many supersmooth tiles the smoothness exponents of the corresponding splines after several convolutions are close to the threshold values (this is so, for example, for $x^2-3x+3$). Table 2.Tiles with equal matrices and different sets of digits: different behaviour of smoothness after convolutions
Remark 4. Consider the polynomial $x^2-x+3$ from Table 1 and its companion matrix. There can be other well-defined sets of digits (lying in different cosets), for example, $D=\{(0,0), (1, 0), (-1, 1)\}$. However, the corresponding tile is not supersmooth. For a comparison of these tiles, see the first two lines of Table 2. The smoothness of the spline of order $\ell$ for the first tile (see the first line of the table) is smaller than $\ell+1/2$, and the smoothness for the second tile is larger. Thus, the presence of supersmoothness depends not only on the matrix, but also on the choice of digits. Another example supporting this observation is considered in Proposition 2. The supersmooth tile considered here is shown in Figure 3.  Figure 3.Supersmooth tiles: (a) the three-digit tile from Remark 4 and (b) the four-digit tile from Remark 6. Proposition 2. Whether a tile is supersmooth depends not only on the matrix, but also on the choice of digits. Proposition 3. There exist two tiles generated by the same matrix (but with different sets of digits) for which the tile $\mathrm B$-splines of order zero have the same smoothness, but tile $\mathrm B$-splines of larger order can have different smoothness. Proof of Propositions 2 and 3. We give a single example for both assertions. Consider the four-digit tiles $G_1$ and $G_2$ generated by the same matrix $M= ((1, -3), (1, 1))$ and the different sets of digits
$$
\begin{equation*}
D_1=\biggl\{\begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 0 \\ -1\end{pmatrix}, \begin{pmatrix} 2 \\ -1\end{pmatrix}, \begin{pmatrix} 1 \\ -1\end{pmatrix}\biggr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_2=\biggl\{\begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 1 \\ 0\end{pmatrix}, \begin{pmatrix} 0 \\ 1\end{pmatrix}, \begin{pmatrix} -1 \\ 1\end{pmatrix}\biggr\}.
\end{equation*}
\notag
$$
Their partitions into four parts are shown in Figure 4.
 Figure 4.Two tiles of equal smoothness generated by the same matrix $M$ (Proposition 3): a supersmooth tile (a) and a nonsupersmooth tile (b). The partition $G= \bigcup_{\boldsymbol{d}\in D} M^{-1}(G+\boldsymbol{d})$ is shown. We denote the symmetrized $\mathrm B$-splines of zero order of these tiles by $B'_0(G_i)$, $i=1, 2$. For each of these splines we have
$$
\begin{equation*}
\Omega=\biggl\{\begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 1 \\ 0\end{pmatrix}, \begin{pmatrix} 0 \\ 1\end{pmatrix}, \begin{pmatrix} 1 \\ 1\end{pmatrix}, \begin{pmatrix} -1 \\ 0\end{pmatrix}, \begin{pmatrix} 0 \\ -1\end{pmatrix}, \begin{pmatrix} -1 \\ -1\end{pmatrix}\biggr\}.
\end{equation*}
\notag
$$
The sets of nonzero coefficients of $B'_0(G_1)$ and $ B'_0(G_2)$ consist of $11$ and 9 numbers, respectively; only the coefficients $c_{(\pm2, 0)}$, $c_{(-1, 1)}$ and $c_{(1, -1)}$ are different. These vectors cannot be represented as $M \omega_i-\omega_j$ for any $\omega_i, \omega_j \in \Omega$, hence the matrices $T$ for these zero-order splines are equal. Namely,
$$
\begin{equation*}
T=\begin{pmatrix} 0.25 & 0 & 1 & 0 & 0 & 0 & 0.5 \\ 0.25 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0.25 \\ 0.5 & 0.25 & 0 & 1 & 0.5 & 0 & 0.25 \\ 0 & 0.5 & 0 & 0 & 0.25 & 1 & 0 \\ 0 & 0.25 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.25 & 0 & 0 \end{pmatrix}.
\end{equation*}
\notag
$$
This, however, is just a coincidence, because in general there is no such agreement for an arbitrary matrix and two sets of digits. The tiles $G_1$ and $G_2$ correspond to the same matrix $T$ and so have the same smoothness. The characteristic polynomial of the matrix $4T$ is $x^7-6x^6+9x^5-4x^4-4x^3+48x+64$. Dividing its zeros by 4, we find the following eigenvalues of $T$:
$$
\begin{equation*}
1, \quad -\frac14, \quad \frac{1 \pm \sqrt{3}}4, \quad\rho= \frac{1+\sqrt[3]{64-3 \sqrt{417}}+\sqrt[3]{64+3 \sqrt{417}}}{12};
\end{equation*}
\notag
$$
we also have two complex zeros of smaller modulus. The eigenvalue 1 and those of modulus 0.5 disappear after the restriction to a subspace, and now the leading eigenvalue is $\rho \approx 0.616876$. The spectral radius of this matrix is $r=2$, so that $\alpha \approx 0.348474$. For the tile $\mathrm B$-splines of order 2 the matrix $T$ is larger, because for $G_1$ the set of nonzero coefficients of the refinement equation for $B'_1(G_1)$ has cardinality 33 and the size of the matrix $T$ is $29 \times 29$, while for $G_2$ we obtain 25 and $27 \times 27$, respectively. In this case, there is no occasional coincidence of the matrices $T$, and the answers are different:
In Table 2, we can see from the third and fourth lines that the difference between smoothness exponents becomes even larger for subsequent convolutions. Unlike $G_2$, the tile $G_1$ gives rise to supersmoothness, which proves Proposition 2. The supersmooth tile $G_1$ enters Table 1 (but for a different matrix and different digits); the corresponding polynomial is $x^2-2x+4$. This proves Propositions 2 and 3. Remark 5. In Table 1 we see an example, for the polynomial $x^2-2x+5$, where two tiles have the same smoothness, convolutions have different smoothness, and both tiles are supersmooth. Using a simple trick we can produce tiles with a greater number of digits from tiles with fewer digits. Proposition 4. Let the matrix $M$ and system of digits $D(M)=\{d_0,\dots,d_{m-1}\}$ generate a tile $G$. Then the matrix $M' = M^2$ and the system of digits $D'(M)=\{Md_i+d_j \mid 0 \leqslant i, j \leqslant m-1\}$ generate the same tile $G$. Proof. The tile $G$ consists of the points of the form
$$
\begin{equation*}
x=M^{-1}\boldsymbol{\Delta}_1+ M^{-2}\boldsymbol{\Delta}_2+\dotsb, \qquad \boldsymbol{\Delta}_i \in D(M).
\end{equation*}
\notag
$$
This can be written as
$$
\begin{equation*}
\begin{gathered} \, x=M^{-2} (M\boldsymbol{\Delta}_1+\boldsymbol{\Delta}_2)+M^{-4} (M\boldsymbol{\Delta}_3+\boldsymbol{\Delta}_4)+\dots=M'^{-1}\boldsymbol{\Delta}'_1+ M'^{-2}\boldsymbol{\Delta}'_2+\dotsb, \\ \boldsymbol{\Delta}'_i \in D'(M). \end{gathered}
\end{equation*}
\notag
$$
Thus, $G$ is a subset of the tile generated by the matrix $M'$ and the system of digits $D'(M)$. The converse inclusion is proved similarly.
For example, taking the two-digit ‘Bear’ tile as $G$, we can obtain it as a four-digit tile with the following matrix and digits:
$$
\begin{equation*}
M'=\begin{pmatrix} -1 & -2 \\ 1 & -2\end{pmatrix}\quad\text{and} \quad D'=\biggl\{\begin{pmatrix} 0 \\ 0\end{pmatrix}, \begin{pmatrix} 1 \\ 0\end{pmatrix}, \begin{pmatrix} 1 \\ 1\end{pmatrix}, \begin{pmatrix} 2 \\ 1\end{pmatrix}\biggr\}.
\end{equation*}
\notag
$$
The four-digit ‘Bear’ tile is shown in Figure 2, b. Of course, the four-digit ‘Bear’ has the same smoothness as the two-digit one, and so we do include it in Table 1 as a separate tile.
This proves the proposition. Remark 6. Considering the same matrix of the four-digit ‘Bear’ with digits $D=\bigl\{ (0,0), (0,-1), (2,-1), (1, -1)\bigr\}$ we obtain another tile (see Figure 3) of lower smoothness. However, the tile $\mathrm B$-splines generated by this tile have a larger smoothness (see the last two lines in Table 2). We obtain three examples (see Table 2) where two tiles are generated by the same matrix and different sets of digits:
Thus, by comparing only the smoothness of two tiles $G_1$ and $G_2$ with equal matrices, one cannot decide which of the $B$-spline is smoother, $B_\ell^{G_1}$ or $B_\ell^{G_2}$. AcknowledgementsThe author is grateful to V. Yu. Protasov for his considerable help and encouragement. The author is also indebted to the developers of the software package [41], which was used in the paper for the construction of tiles.
Citation in format AMSBIB
\Bibitem{Zai25} |
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