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Convergence of regularized greedy approximations I. P. Svetlov Lomonosov Moscow State University
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    A greedy algorithm is characterized among all non-linear approximation methods by the presence of the greedy step principle, which is based on maximization of the functional determined from information from the previous steps of the algorithm. Non-linear approximation is widely used in areas such as machine learning theory and compressed sensing. In this paper, we study greedy approximation of an element of a real Banach space. We first define the ordinary greedy algorithm for a biorthogonal system [1]. Let $X$ be a separable real Banach space of real numbers and let $\Psi=\{\psi_k\}_{k\in I}$ be a complete minimal system in $X$, where $I$ is a countable set. If the system is complete and minimal, we can uniquely define a biorthogonal system $\{\psi_k , \psi^*_j\}$, $\psi_k \in X$, $\psi^*_j \in X^*$, $\psi^*_j(\psi_k) = \delta^j_k$. Let the elements of the conjugate system satisfy the additional condition $\sup_{j}\|\psi^*_j\|<\infty$, which will be useful below. Consider the formal expansion of an element $f \in X$ in $\Psi$:
$$
\begin{equation*}
f \sim \sum_{k \in I} \xi_k \psi_k, \qquad \xi_k = \psi^*_k (f).
\end{equation*}
\notag
$$
The condition $\sup_j\|\psi^*_j\|<\infty$is sufficient for $\xi_k \to 0$ as $k \to \infty$. As a result, we can define a permutation $\rho,$ of the set of indices corresponding to the non-zero coefficients in the expansion of $f$ so as to have
$$
\begin{equation*}
|\xi_{k_1}(f,\Psi)|\geqslant|\xi_{k_2}(f,\Psi)|\geqslant|\xi_{k_3}(f,\Psi)| \geqslant\cdots,
\end{equation*}
\notag
$$
where $\rho(j) = k_j$, $j = 1, 2, 3, \dots$ . Let $D(f)$ be the set of such permutations. If all the above inequalities are strict, then $D(f)$ consists of only a single permutation. Consider the $m$-terms greedy approximant of $f$ defined by
$$
\begin{equation*}
G_m(f)=G_m(f,\Psi, \rho)=\sum_{j=1}^m \xi_{k_j}\psi_{k_j}.
\end{equation*}
\notag
$$
We set It is clear that $\theta_m \to 0$ as $m\to \infty$.
If the expansion of $f$ if $\Psi$ has only finite number of non-zero coefficients, then the algorithm converges in a finite number of steps, and this case is devoid of interest. So, in what follows, we assume that the expansion of $f$ in $\Psi$ is infinite. In order to show how the convergence of the new version of the greedy algorithm differs from that for the ordinary greedy algorithm, we need the following definition from [2]. Definition 1. A biorthogonal system $\{\psi_k , \psi^*_j\}$ is called quasi-greedy if, for every $f\in X$ and each permutation $\rho \in D(f)$, where $C$ is independent of $f$, $\rho$ and $m$. From this definition it follows that every unconditional basis (for example, the Haar system) is a quasi-greedy basis in $L_p(0, 1)$, $1<p<\infty$. Wojtaszczyk [3] proved the following theorem Theorem 1. A biorthogonal system $\{\psi_k , \psi^*_j\}$ is quasi-greedy if and only if, for any $f \in X$ and permutation $\rho \in D(f)$, However, if the system is not quasi-greedy, then the ordinary greedy algorithm does not always converge to the original element even in reflexive spaces [4]. For example, for the trigonometric system in $L_p$, $p\in [1, 2)\cup(2, \infty]$, there exists an element $f$ such that $G_m(f)$ does not converge to $f.$ In this paper we discuss a new type of a greedy algorithm, which will be called the regularized greedy algorithm (REGGA). We prove the convergence of the REGGA in Efimov–Stechkin spaces. In 1961, Efimov and Stechkin [5] considered the reflexive spaces in which the weak convergence of sequences on the unit sphere coincides with the strong convergence. The name Efimov–Stechkin space for this class of spaces was introduced by Singer [6]. We can give a slightly different definition of Efimov–Stechkin spaces. For this purpose, consider the following property, known as the Kadec–Klee property: if a sequence $\{x_n\} \in X$ converges weakly to some $x \in X$ and $\lim _{n \to \infty}\|x_n\| = \|x\|$, then $\{x_n\}$ converges strongly to $x.$ A Banach space $X$ is a Efimov–Stechkin space if it is reflexive and has the Kadec–Klee property [7]. Such spaces, which were also considered in the context of greedy approximations (see [8]), proved quite useful in approximation theory, in variational problems, and in problems of convergence of projection methods [9]. We will construct the REGGA algorithm based on the ordinary greedy algorithm by expanding an element $f$ of a Efimov–Stechkin space in a biorthogonal system. The principal feature of the REGGA is that the REGGA is constructed by subtracting elements from $G_m(f)$ э the $m$-term greedy approximant whose coefficients in the expansion in the biorthogonal system do not exceed in absolute value the coefficients of the $m$-term greedy approximant. § 1. Regularized approximationIn this section, we construct a regularized greedy approximation based on the greedy algorithm for a biorthogonal system. This approximation is obtained from the ordinary greedy approximation as follows. Let $X$ be a separable real Banach space and $\{\psi_k , \psi^*_j\}$, $\psi_k \in X$, $\psi^*_j \in X^*$, $\psi^*_j(\psi_k) = \delta^j_k$, $\sup_{j}\|\psi^*_j\|<\infty$. be a biorthogonal system. We fix three null sequences $\{\beta_m\}$, $\{\beta_m'\}$, $\{\varepsilon_m\}$ of non-negative numbers independent of $f$. Consider an element $f \in X$, $f\ne 0$, and define the set $\mathcal A_m=\mathcal A_m(G_m(f))\in X$ consisting of elements $g = \sum _{j=1}^\infty \varkappa_j \psi_j$ whose coefficients satisfy
$$
\begin{equation}
|\varkappa_j| \leqslant \beta_m, \qquad j=j_1,\dots, j_m,
\end{equation}
\tag{1.1}
$$
$$
\begin{equation}
|\varkappa_j| \leqslant \beta_m' + \theta_m, \qquad j\ne j_1,\dots, j_m,
\end{equation}
\tag{1.2}
$$
where $j_1, \dots, j_m$ are the indices of the corresponding coefficients of $G_m(f).$ Then we take $g_m \in \mathcal A_m$ satisfying
$$
\begin{equation}
\|G_m - {g}_m \| \leqslant \inf_{g \in \mathcal A_m} \|G_m - g\| + \varepsilon_m.
\end{equation}
\tag{1.3}
$$
We will sometimes assume that $\varepsilon_m>0$ to ensure that the required element $g_m $ exists. So, we have defined the regularized greedy approximation that depends on $\beta_m$, $\beta_m'$, and $\varepsilon_m$. We denote this approximation by the REGGA($f$, $\{\beta_m\}$, $\{\beta_m'\}$, $\{\varepsilon_m\}$). We set $G_m - g_m = h_m$. The main goal of this algorithm is to make sure that, at each step $m$, from the known element $G_m(f)$ and its expansion to construct a system of elements $\{h_m\}$ converging to the desired function $f$. As mentioned above, if we take only $G_m(f) $, then convergence is not guaranteed, for example, since the $m$-term greedy approximants can be unbounded. So, in the construction of the REGGA, we make sure that the elements of $h_m$ will be bounded. This is achieved by taking an element of the near-best approximation to $G_m(f)$ at each step (see (1.3)). § 2. Main resultsWe recall some definitions. Definition 2. A separable real Banach space $X$ is called a Efimov–Stechkin space if it is reflexive and has the $A$-property: if $x\in S$, $x_n \in S$, $x_n \xrightarrow {w} x$, then $x_n \to x$. Here, $S=\{x\in X\colon \|x\|=1\}$. Definition 3. A set is called boundedly weakly compact if its intersection with every closed ball is weakly compact [10]. We now us formulate the main results on convergence of regularized greedy approximations. Theorem 2. Let $X$ be a separable reflexive space, let $\Psi=\{\psi_k\}$ be a complete and minimal system in $X$, and let the linear span of the functionals of the biorthogonal system forms is dense in the dual space. Then, for all null sequences $\beta_m$, $\beta_m'$ and $\varepsilon_m$ of non-negative numbers and any function $f$, the sequence of elements $\{h_m\}$ converges weakly to $f$. Theorem 3. Let $X$ be a Efimov–Stechkin space and let the linear span of functionals of a biorthogonal system is dense in the dual space. Then, for all null sequences $\beta_m$, $\beta_m'$ and $\varepsilon_m$ of non-negative numbers and any function $f$, the sequence of elements $\{h_m\}$ converges strongly to the original element $f$. Proof of Theorem 2. The set $\mathcal A_m$ is convex and closed. Hence it is boundedly weakly compact set, since $X$ is reflexive. Any non-empty weakly boundedly compact set is a set of existence. If a sequence converges weakly, then this sequence is bounded, and, for a system from the dual space whose linear span is dense in this space, the sequence of elements obtained by applying any element of this system to the sequence in the original space should converge to some limit. By definition, the norms of elements of the sequence $\{h_m\}$ are bounded from above: It remains to choose an appropriate system in the dual space. As such a system, we can take the system $(\psi^*_j)$, which is dual to $\{ \psi_j, \psi^*_n \}$. By ${\hat h}_{m,j}$ we denote the coefficients in the expansion of $h_m$ in $\Psi$.
Case 1: $\xi_j\neq 0$. There exists $m_0$ such that $\theta_{m_0}<|\xi_j|$, because $\theta_m\to 0$ as $m\to\infty$. For $m\geqslant m_0$, we have $j\in\{j_1,\dots,j_m\}$. By definition of the set $\mathcal A_m$, $|\widehat h_{m,j} - \xi_j| \leqslant \beta_m$. Since $\lim_{m\to\infty} \beta_m = 0$, we have $\widehat h_{m,j}\to \xi_j$ as $m\to\infty$. Case 2: $\xi_j = 0$. For any $m$ we have $\theta_m \geqslant |\xi_j|$. Hence $j\notin\{j_1,\dots,j_m\}$. By definition of the set $\mathcal A_m$, $|\widehat h_{m,j} - \xi_j| \leqslant \beta_m' + \theta_m$. Since $\lim_{m\to\infty} \beta_m' = 0$ and since $\lim_{m\to\infty} \theta_m = 0$, we have $\widehat h_{m,j}\to \xi_j$ as $m\to\infty$. One thing is worth pointing out: since $\xi_j= 0$, the corresponding coefficients in the expansions $G_m(f)$ and $f$ are zero. Hence $\{h_m\}$ converges weakly to the original element. Theorem 2 is proved. Let us mow prove Theorem 3. Proof of Theorem 3. To begin, we note that If this were not so, then there would exist $\{m_j\}$ such that $\lim_{j \to \infty}\|{h}_{m_j}\| \ne \|f\|$. There are two cases to consider.
Case 1: $\lim_{j \to \infty}\|h_{m_j}\| = \gamma < \|f\|$. Then $\|w- \lim_{j \to \infty}h_{m_j}\|\leqslant \gamma <\|f\|$ by (2.1). But the sequence $\{h_m\}$ converges weakly to $f$, which contradicts the inequality $\lim_{j \to \infty}\|h_{m_j}\| < \|f\|$. Case 2: $\lim_{j \to \infty}\|h_{m_j}\| = \gamma \geqslant \|f\|$. This case is impossible because of (2.1). We can normalize the terms of the sequence $\{h_m\}$ such that ${\widetilde h_m}=h_m/\|h_m\|$; we can also assume that $\widetilde f = f/\|f\|$. Next, $\{h_m\}$ converges weakly to $f$, and so in view of (2.2) we can apply the $A$-property to $\{\widetilde h_m\}$ and $\widetilde f$. Hence $\{h_m\}$ converges strongly to $f$. Theorem 3 is proved. § 3. The convergence to zero of the coefficients $\beta_m$, $\beta_m'$ is essentialThe conditions from Theorem 3 that the sequences $\{\beta_m\}$ and $\{\beta_m'\}$ tend to zero used are essential. If condition (1.1) is violated, we can take $g_m=G_m$ at the $m$th step. Then $h_m=0$, and there is no convergence to the initial element $f$ if $f\ne 0$. Let us show that the condition $\beta_m^\prime \to 0$ in (1.2) cannot also be abandoned. Let $X=L_p$, $p\geqslant 1$. Consider the function
$$
\begin{equation*}
f(x) = \begin{cases} n &\text{if }x\in [-\pi, -\alpha] \cup [\alpha, \pi], \\ k &\text{if }x \in [-\alpha, \alpha], \end{cases}
\end{equation*}
\notag
$$
where $0<\alpha<\pi$, $\alpha\ne \pi/2$. Let us find conditions on $\alpha$, $k$, $n$ under which We have
$$
\begin{equation*}
\frac1{2\pi} \biggl[\int_{-\pi}^{-\alpha} n \, dx + \int_{-\alpha}^{\alpha} k \, dx + \int_{\alpha}^{\pi} n \, dx \biggr] = \frac1{2\pi} [2k\alpha + 2n\pi - 2n\alpha ] = 0.
\end{equation*}
\notag
$$
So, choosing $n>0$, $k<0$, we have $\pi/\alpha = 1 - k/n$. Recall the criterion of for an element of best approximation from a subspace $M$ in $L_p$, $p\geqslant1$:
$$
\begin{equation}
\int_a^b |x(t)-y |^{p-1}\operatorname{sign}(x(t)-y)y\, dt =0 \quad \forall\, y \in M.
\end{equation}
\tag{3.1}
$$
We will approximate the function by constants. Suppose that 0 is a constant of best approximation for $f$ for $p\geqslant1, p\ne2.$ Substituting $|k| = \frac{\pi-\alpha}{\alpha}|n|$, $y^*\ne0$ into equation (3.1), we obtain
$$
\begin{equation*}
-2\alpha |k|^{p-1} + 2(\pi-\alpha) |n|^{p-1} = 2(\pi-\alpha) |n|^{p-1}\biggl(1 - \biggl(\frac{\pi-\alpha}{\alpha}\biggr)^{p-2}\biggr).
\end{equation*}
\notag
$$
But this number is not zero, because Hence 0 is not an element of best approximation for $f$ in any $L_p$, $p\geqslant1$, $p\ne2$. Now consider the space $L_2[-\pi, \pi].$ The criterion of for an element of best approximation of $f$ by constants in $L_2$ is as follows: As a result, 0 is an element of best approximation for $f$ in $L_2$, so the REGGA converges to the original function only in $L_2.$Assume that $f$ has an element of best approximation $c_0\neq 0$. Consider $\beta_m^\prime \geqslant |c_0|$ and take ${\tilde g_m}=G_m - (f-c_0)$. This function lies in $\mathcal A_m$ because the largest coefficients in its expansion are cancelled, and there remain only $\varkappa_j, j\ne0$ that satisfy condition (1.2). For $c_0$, we have $c_0 \leqslant {\beta_m^\prime}$. Next, $G_m-{\tilde g_m}={\tilde h_m}$. Hence $\|\widetilde h_m\|_p=\|f-c_0\|_p < \|f\|_p $. If we choose $g_m$ from condition (1.3), then $\|G_m-g_m\|_p\leqslant \|G_m-\widetilde g_m\|+\varepsilon_m = \|f-c_0\|_p +\varepsilon_m < \|f\|_p$. Now, setting $h_m=G_m-g_m$, we find that $h_m$ does not converge to $f$. § 4. Divergence in $\widetilde C[-\pi, \pi]$Consider the real space $X=\widetilde C[-\pi, \pi]$ of continuous functions such that $f(-\pi)=f(\pi)$. A function from this space can be continued to the entire line as a $2\pi$-periodic continuous function. In $X$, the trigonometric system in real form is considered as a system $\Psi=\{\psi_k\}_{k\in I}$.Consider a function $f\in\widetilde C[-\pi, \pi]$ whose expansion has infinitely many non-zero Fourier coefficients. Let the Fourier series of $f$, in which the terms of the series are rearranged in the order of non-increasing absolute value of the coefficients, have the form Let $\beta_m^\prime = 0$ for any natural $m$, and let $\beta_m$ be positive and tend to 0. Our divergence result is as follows. Proposition 1. Let $f \in \widetilde C[-\pi, \pi]$ be a not trigonometric polynomial, and let $\varepsilon_m$ be an arbitrary positive null sequence, $\beta_m' = 0$, and $\beta_m = \theta_m$. Then some realization of the REGGA algorithm does not converge to $f$. Note that $\theta_m$ is defined from $f$ by (*). Of the two terms $\xi_{j_1}\psi_{j_1}$ and $\xi_{j_2}\psi_{j_2}$ in the expansion of $f$, at least one is of the form $\xi \cos(jx)$ or $\xi \sin(jx),$ $\xi \ne 0$, $j \ne 0$. Without loss of generality, we may assume that this term is $\xi\cos(jx);$ (the case $\xi\sin(jx),$ is considered similarly). We will show that, for sufficiently large $m$, there exists $\alpha > 0$ such that
$$
\begin{equation}
\|G_m - g\|_C \geqslant \alpha, \qquad g\in \mathcal A_m.
\end{equation}
\tag{4.1}
$$
We recall that $\theta_m=\min ( |\xi_{j_1}|,\dots, |\xi_{j_m}| )$. For sufficiently large $m$, we have $\beta_m \leqslant |\xi|/2$. The trigonometric expansion of any $g \in \mathcal A_m$ involves $\cos(jx)$ with some coefficient $\gamma$, which satisfies $|\gamma|\leqslant \beta_m\leqslant |\xi|/2$. Hence the expansion of $G_m-g$ contains the term $(\xi- \gamma)\cos(jx)$, and $|\xi- \gamma|\geqslant |\xi|-|\gamma|\geqslant |\xi|/2$. We know that
$$
\begin{equation*}
\xi - \gamma = \frac1{\pi}\int_{-\pi}^\pi (G_m(x)-g(x))\cos(jx) \, dx,
\end{equation*}
\notag
$$
and hence
$$
\begin{equation*}
\begin{aligned} \, |\xi - \gamma| &= \biggl|\frac1{\pi}\int_{-\pi}^\pi (G_m(x)-g(x))\cos(jx) \, dx\biggr|\leqslant \frac1{\pi}\int_{-\pi}^\pi |(G_m(x)-g(x))\cos(jx)| \, dx \\ &\leqslant \frac1{\pi}\int_{-\pi}^\pi |(G_m(x)-g(x))| \, dx \leqslant 2\|G_m-g\|_C. \end{aligned}
\end{equation*}
\notag
$$
We can now put $\alpha = |\xi|/4$. Consider an element $g_m\in \mathcal A_m$ that satisfies
$$
\begin{equation}
\|G_m-g_m\|_C \leqslant \inf_{g\in A_m}\|G_m-g\|_C + \frac{\varepsilon_m}2
\end{equation}
\tag{4.2}
$$
(this inequality is stronger than (1.3)). The set $\mathcal A_m$ consists of all continuous functions whose Fourier coefficients satisfy conditions (1.1) and (1.2). Our aim is to make the coefficients even smaller in order to be able to vary them. To do this, we take a small $\delta_m>0$, and define We have
$$
\begin{equation}
\|G_m-g_m^{(1)}\|_C \leqslant \inf_{g\in \mathcal A_m}\|G_m-g\|_C + \varepsilon_m.
\end{equation}
\tag{4.3}
$$
Here, $g_m^{(1)}\in \mathcal A_m$, and, in addition, the absolute values of the Fourier coefficients of $g_m^{(1)}$ for $\{\psi_{j_1}, \dots, \psi_{j_m}\}$ are majorized by $(1-\delta_m)\beta_m$, and, for the remaining elements of the system $\Psi$, they are majorized by $(1-\delta_m)\theta_m$. We now set $g_m^{(2)} = g_m^{(1)}+h$. As a function $h$ we take a continuous function on $[-\pi,\pi]$ which vanishes everywhere except as small neighbourhood $(-\delta_2,\delta_2)$ of zero. In this neighbourhood, this function keeps the same sign, and its absolute values attains its maximum $\alpha/2$ at zero. By choosing a sufficiently small $\delta_2$, we can make the Fourier coefficients of the function $h$ small enough, which gives us that $g_m^{(2)} \in \mathcal A_m$. If $ (G_m-g_m^{(1)})(0)\geqslant0$, we set $h(0) = \alpha/2$, and if $(G_m-g_m^{(1)})(0)<0$, we define $h(0) = -\alpha/2$. We need to check that
$$
\begin{equation}
\|G_m - g_m^{(2)}\|_C \leqslant \|G_m - g_m^{(1)}\|_C.
\end{equation}
\tag{4.4}
$$
Consider the case $(G_m-g_m^{(1)})(0)\geqslant0$; the arguments in the second case are similar. In the case $x\in[-\pi,-\delta_2]\cup[\delta_2,\pi]$, we have $h(x)=0$, and
$$
\begin{equation*}
|G_m(x) - g_m^{(2)}(x)| = |G_m(x) - g_m^{(1)}(x)| \leqslant \|G_m - g_m^{(1)}\|_C.
\end{equation*}
\notag
$$
Now let $x\in(-\delta_2,\delta_2)$. Hence $h(x)>0$, and so
$$
\begin{equation*}
G_m(x) - g_m^{(2)}(x) = G_m(x) - g_m^{(1)}(x) - h(x) < \|G_m - g_m^{(1)}\|_C.
\end{equation*}
\notag
$$
On the other hand, if $\delta_2$ is a sufficiently small, then, for all $x\in(-\delta_2,\delta_2)$, whence
$$
\begin{equation*}
(G_m(x)-g_m^{(2)})(x) \geqslant -\frac{\alpha}2 - h(x) \geqslant -\alpha \geqslant - \|G_m - g_m^{(1)}\|_C.
\end{equation*}
\notag
$$
So, for all $x\in[-\pi,\pi]$,
$$
\begin{equation*}
|G_m(x) - g_m^{(2)}(x)| \leqslant \|G_m - g_m^{(1)}\|_C,
\end{equation*}
\notag
$$
which implies (4.4). The sequences $G_m-g_m^{(1)}$, $G_m-g_m^{(2)}$ cannot both simultaneously converge to $f$ even weakly, since their difference $h_m(x)$ at zero is $\alpha/2$ for any $m$. § 5. Divergence in $L_1[-\pi, \pi]$Now we consider the real space $X=L_1[-\pi, \pi].$ As in the case $\widetilde C[-\pi, \pi]$, we take as $\Psi = \{\psi_k\}_{k\in I}$ the trigonometric system in the real form
$$
\begin{equation*}
\{\cos(jx)\}_{j\geqslant 0} \cup \{\sin(jx)\}_{j > 0}.
\end{equation*}
\notag
$$
Let $f\in L_1[-\pi, \pi]$. Assume that its expansion has infinitely many non-zero Fourier coefficients. Let the Fourier series of $f$, in which the terms of are arranged in the order of non-increasing absolute values of the coefficients, have the form Let $\beta_m^\prime = 0$ for any natural $m$, and let $\beta_m$ be positive and tend to 0.The corresponding divergence result is as follows. Proposition 2. Let $f \in L_1[-\pi, \pi]$ be a not trigonometric polynomial, and let $\varepsilon_m$ be an arbitrary positive mull sequence, $\beta_m' = 0$, $\beta_m = \theta_m$. Then there exists an implementation of the REGGA algorithm that does not converge to $f$. Note that $\theta_m$ is defined from $f$ by (*). Of the two terms $\xi_{j_1}\psi_{j_1}$ and $\xi_{j_2}\psi_{j_2}$ in the expansion of $f$, at least one is of the form $\xi \cos(jx)$ or $\xi \sin(jx),$ $\xi \ne 0$, $j \ne 0$. Without loss of generality, we may assume that this term is $\xi\cos(jx)$ (the case $\xi\sin(jx),$ is considered similarly). Recall that $\theta_m=\min_{i=1, \dots, m} |\xi_{j_i}|$. For sufficiently large $m$, we have $\beta_m \leqslant |\xi|/2$. The trigonometric expansion of any $g \in \mathcal A_m$ involves $\cos(jx)$ with some coefficient $\gamma$, which satisfies $|\gamma|\leqslant \beta_m\leqslant |\xi|/2$. The expansion of $G_m-g$ has the term $(\xi- \gamma)\cos(jx)$, and $|\xi- \gamma|\geqslant |\xi|- |\gamma|\geqslant |\xi|/2$. We know that
$$
\begin{equation*}
\xi - \gamma = \frac1{\pi}\int_{-\pi}^\pi (G_m(x)-g(x))\cos(jx) \, dx.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, |\xi - \gamma| &= \biggl|\frac1{\pi}\int_{-\pi}^\pi (G_m(x)-g(x))\cos(jx) \, dx\biggr|\leqslant \frac1{\pi}\int_{-\pi}^\pi |(G_m(x)-g(x))\cos(jx)| \, dx \\ &\leqslant\frac1{\pi}\int_{-\pi}^\pi |(G_m(x)-g(x))| \, dx \leqslant \frac1\pi \|G_m-g\|_{L_1}. \end{aligned}
\end{equation*}
\notag
$$
We set $\alpha = \pi |\xi|/2$. In $L_1[-\pi, \pi]$, an analogue of(4.1) holds, and $h_m$ are also separated from zero, that is, For a small $\delta_m>0$, we set We have
$$
\begin{equation}
\|G_m-g_m^{(1)}\|_{L_1} \leqslant \inf_{g\in \mathcal A_m}\|G_m-g\|_{L_1} + \varepsilon_m.
\end{equation}
\tag{5.1}
$$
Next, $g_m^{(1)}\in \mathcal A_m$, and, in addition, the absolute values of the Fourier coefficients of the function $g_m^{(1)}$ for $\{\psi_{j_1}, \dots, \psi_{j_m}\}$ are majorized by $(1-\delta_m)\beta_m$, and the remaining coefficients are majorized by $(1-\delta_m)\theta_m$. We now set $h_m = G_m - g_m^{(1)}$. Recall that the Rudin–Shapiro polynomial is given by $P_n(z)=\sum_{j=0}^{2^n-1} \sigma_j z^j$, $\sigma_j\in\{1,-1\}$. An important property of the polynomials $P_n(z)$ is that $|P_n(z)|\leqslant \sqrt{2^{n+1}}$ for $|z|=1$ (see [12]). Our main aim in the proof of the proposition is to construct functions $\widetilde h_m$ such that $\|\widetilde h_m - h_m\|_{L_1}=\|h_m\|_{L_1}$, $\|\widetilde h_m\|_{L_1}\leqslant\|h_m\|_{L_1} $, and the absolute values of all Fourier coefficients of the function $\widetilde h_m$ for $\{\psi_{j_1}, \ldots, \psi_{j_m}\}$ do not exceed $\beta_m$, and the remaining coefficients of $\widetilde h_m$ are majorized by $\theta_m.$ To satisfy the last condition, it is sufficient that the absolute values of all Fourier coefficients of $\widetilde h_m - h_m$ should not exceed $\delta_m.$ We set $\widetilde g_m = G_m -\widetilde h_m$. Both sequences $\{g_m\}$ and $\{\widetilde g_m\}$ satisfy conditions (1.1) and (1.2). The sequences $\{h_m\}$ and $\{\widetilde h_m\}$ cannot both converge to $f$ since $\|\widetilde h_m - h_m\|_{L_1}=\|h_m\|_{L_1}\geqslant c > 0$. The following lemma is crucial for the construction of the functions $\widetilde h_m$. Lemma 1. For any function $h\in L_1$ and any $\delta>0$, there exists a function $\sigma$ with values $1$, $-1$ such that the absolute values of all Fourier coefficients of the function $h\sigma$ do not exceed $\delta$. We apply this lemma to the function $h=h_m$ and the number $\delta=\delta_m$. Since
$$
\begin{equation*}
\|h_m+h_m\sigma\|_{L_1} + \|h_m-h_m\sigma\|_{L_1} = 2\|h_m\|_{L_1},
\end{equation*}
\notag
$$
one of the functions $\widetilde h_m=h_m+h_m+h_m\sigma$, $\widetilde h_m=h_m-h_m\sigma$ satisfies $\|\widetilde h_m\|_{L_1} \leqslant \|h_m\|_{L_1}$. We take this function. Everywhere, we will use the Fourier coefficients in real form. But to estimate such coefficients, it suffices to estimate the coefficients in complex form. In addition, in the proof of Lemma 2 and other lemmas, it will be more convenient to consider the coefficients in complex form. The main role in the proof of Lemma 1 is played by the following lemma. Lemma 2. For any $\theta>0$, there exists a function $\sigma$ with values $1$, $-1$ such that the absolute values of all Fourier coefficients of the function $\sigma$ do not exceed $\theta$. We will derive Lemma 1 from Lemma 2. Let $h$ be a trigonometric polynomial: The function $\sigma$ constructed in Lemma 2 has the Fourier series expansion
$$
\begin{equation*}
\sigma(x)=\sum_{k=-\infty}^{\infty} \widehat \sigma(k) e^{ikx},
\end{equation*}
\notag
$$
where $|\widehat \sigma(k)|\leqslant \theta$, $k\in\mathbb Z$. The function $h(x)\sigma(x)$ expands in a Fourier as where
$$
\begin{equation*}
c_k=\sum_{l=-n}^{n} \widehat h(l) \widehat \sigma(k-l).
\end{equation*}
\notag
$$
We have
$$
\begin{equation*}
|c_k|\leqslant \sum_{l=-n}^{n}| \widehat h(l) |\, |\widehat \sigma(k-l)|\leqslant \sum_{l=-n}^{n}| \widehat h(l) |\theta.
\end{equation*}
\notag
$$
For
$$
\begin{equation*}
\theta = \delta \biggl(\sum_{l=-n}^{n}| \widehat h(l) | \biggr)^{-1},
\end{equation*}
\notag
$$
we see that $|c_k|\leqslant \delta$ for all $k$. In the general case, where $h$ is not a trigonometric polynomial, $h$ can be approximated by a trigonometric polynomial $T(h)$ so that $\|h- T(h)\|_{L_1}\leqslant \delta/2$. By the above, there exists a function $\sigma$ with values $1$, $-1$ such that all Fourier coefficients of the function $T(h) \sigma$ do not exceed $\delta/2$ in absolute value. We have
$$
\begin{equation*}
\|h\sigma - T(h) \sigma\|_{L_1} = \|h-T(h)\|_{L_1}\leqslant \frac{\delta}2.
\end{equation*}
\notag
$$
Then all Fourier coefficients of the function $|h\sigma - T( h) \sigma|$ do not exceed $ \delta/2$ in absolute value. Hence, the absolute values of the Fourier coefficients of the function $h\sigma$ do not exceed $\delta.$ To prove Lemma 2, we use the Rudin–Shapiro polynomials $P_{n}(z) = \sum_{j=0}^{2^n-1} \sigma_j z^j$. We split $[0,2\pi)$ into $2^n$ into $2^n$ equal half-open intervals $\Delta_0,\dots, \Delta_{2^n-1}$. We denote $\Delta_{j}$ by $[b_j, b_{j+1})$, where $b_j = j \pi/2^{n-1}$. On the $j$th half-open interval, we define the function to be equal to $\sigma_j$. We claim that this function satisfies the conditions of Lemma 2. Proof of Lemma 2. The $k$th coefficient is equal to Expanding, we have two sums $S_1=\sigma_0 e^{(ikb_0)} + \dots + \sigma_{2^n - 1} e^{(ikb_{2^n - 1})}$ and $S_2=\sigma_0 e^{(ikb_1)} + \dots + \sigma_{2^n - 1}e^{(ikb_{2^n})}$. Hence $S_1=P_{n}(e^{(ikb_1)})$ and $ S_2=e^{(ikb_1)}S_1$. Let us estimate $c_k(\sigma)$ from above. There are two cases to consider: 1) $|k|\geqslant 2^n$; 2) $|k|<2^n$.
Case 1). By properties of Rudin–Shapiro polynomials,
$$
\begin{equation*}
|c_k(\sigma)| = \biggl|\frac{i}k(S_1 - S_2)\biggr|\leqslant \frac1k(|S_1|+|S_2|)\leqslant \frac1k\bigl(\sqrt{2^{n+1}}+\sqrt{2^{n+1}}\,\bigr).
\end{equation*}
\notag
$$
Hence
Case 2). We rewrite $S_1-S_2$ as $S_1-e^{(ikb_1)}S_1$. Transforming the difference of trigonometric functions in a product, we have
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\frac{i}k(S_1 - e^{(ikb_1)}S_1)\biggr| \leqslant \biggl|\frac{i}k\biggr|\, \biggl|\biggl( 2\sin^2\biggl(\frac{kb_1}{2}\biggr) - i\sin({kb_1})\biggr)S_1\biggr| \\ &\qquad\leqslant 2^{3/2}\biggl|\frac{i}k\biggr|\, \biggl|\sin\biggl(\frac{kb_1}{2}\biggr) \sin\biggl(\frac{kb_1}{2} - \frac{\pi}{4}\biggr)S_1\biggr| \leqslant 2^{3/2}\biggl|\frac1k \biggl(\frac{kb_1}{2}\biggr)S_1\biggr| = |b_1S_1| \leqslant \frac{4\pi }{2^{n/2}}. \end{aligned}
\end{equation*}
\notag
$$
Combining two cases for the index $k$, we get the upper estimate Lemma 2 is proved. § 6. Divergence in reflexive non-Efimov–Stechkin spaceLet us show that in Theorem 3 one cannot replace the Efimov–Stechkin spaces by more general reflexive spaces. To this end, consider the space of sequences with the norm $\|(a_1, a_2, \dots)\| = \max(\|(a_1, a_2, \dots)\|_{l_2}, 2|a_1|)$. Consider the vector $u=(u_k)$, where $u_k=2^{-k}$. It is clear that $\|u\|=1$. Then, in the space of sequences just defined, and under the condition $\beta_m=\beta_m'=\theta_m$, the REGGA will not always converge to $u$. It is easily seen that $G_m(u) = (1/2, 1/4, \dots, 1/2^m, 0, \dots)$. Let us show that we can choose $g_m \in \mathcal A_m(G_m(u))$ so as to have $\|g_m\|=1/\sqrt2$ for $m>2$. For this purpose, it suffices to take
$$
\begin{equation*}
g_m =\biggl(\underbrace{\frac1{2^m},\dots,\frac1{2^m}}_{2^{2m-1}}, 0, \dots\biggr).
\end{equation*}
\notag
$$
Let us estimate The first coefficient of $\xi_{j_1}$ in the expansion of $G_m$ is always $1/2$, and, for the first coefficient in the expansion of $g$, we have $|\varkappa_1|\leqslant 1/2^m$. Hence
$$
\begin{equation*}
\|G_m - g\| \geqslant 2(G_m - g)_1 \geqslant 2\biggl(\frac12-\frac1{2^m}\biggr) = 1- \frac1{2^{m-1}}.
\end{equation*}
\notag
$$
We recall that $h_m = G_m - g_m$. Let us estimate $\|h_m\|$ for $m>3$. To this end, we use the explicit formula
$$
\begin{equation*}
\!h_m \,{=}\, \bigl(\widehat h_{m,1}, \widehat h_{m,2}, \dots\bigr) \,{=}\, \biggl(\frac1{2}-\frac1{2^m}, \dots, \frac1{2^{m-1}} - \frac1{2^m}, 0, -\frac1{2^m}, -\frac1{2^m}, \dots, -\frac1{2^m}, 0, \dots\biggr).
\end{equation*}
\notag
$$
Considering the first coordinate $\widehat h_{m,1}$ of the vector $h_m$, we have $2|\widehat h_{m,1}|=1- 1/2^{m-1}$. Next, we have
$$
\begin{equation*}
\|h_m\|_{l_2}= \frac{\sqrt{(2^{2m-1}-m)+\sum_{i=1}^{m-1}(2^i - 1)^2}}{2^m}.
\end{equation*}
\notag
$$
As a result, $\|h_m\|_{l_2} \leqslant \sqrt{5/6}$. Hence $\|h_m\| = 1- 1/2^{m-1}$. We estimate $\|u - h_m\|$ for some sufficiently large $m$, We have
$$
\begin{equation*}
u - h_m = \biggl(\underbrace{\frac1{2^m}, \dots, \frac1{2^m}}_{m}, \underbrace{\frac{3}{2^{m+1}}, \frac{5}{2^{m+2}}, \dots, \frac{2^j + 1}{2^{m+j}}, \dots}_{2^{2m-1}-m}\,, \frac1{2^{2^{2m-1}+1}}, \frac1{2^{2^{2m-1}+2}}, \dots\biggr).
\end{equation*}
\notag
$$
Let us estimate $\|u - h_m\|_{l_2}$. We have
$$
\begin{equation*}
\begin{aligned} \, \|u - h_m\|_{l_2} &= \frac1{2^m} \sqrt{m+\sum_{i=1}^{2^{2m-1} - m} \frac{(2^i + 1)^2}{2^{2i}} + \sum_{j=1}^{\infty} \frac{2^{2m}}{2^{2^{2m}+2j}}} \\ &=\frac1{2^m} \sqrt{2^{2m-1}+\sum_{i=1}^{2^{2m-1} - m} (\frac1{2^{i-1}} + \frac1{2^{2i}}) + \sum_{j=1}^{\infty} \frac{2^{2m}}{2^{2^{2m}+2j}}}. \end{aligned}
\end{equation*}
\notag
$$
Hence $\|u-h_m\|_{l_2}>1/2$. As a result, $\varlimsup_{m\to \infty} \|u-h_m\|\geqslant 1/2$. This shows that $h_m$ does not converge to $u$.
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\Bibitem{Sve25} |
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