| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Brief communications On Fourier series in the multiple trigonometric system M. G. Grigoryana, S. V. Konyaginb a Yerevan State University b Lomonosov Moscow State University
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    This is a continuation of [1]–[5]. We introduce the requisite notation and definitions: $\mathbb{T}^{d}:=[{-\pi},{\pi}]^{d}$; $\mathbf{x}=(x_{1},\dots,x_{d})\in \mathbb{R}^{d}$, $\mathbf{k}=(k_{1},\dots,k_{d})\in \mathbb{Z}^{d}$, where $d\in\mathbb{N}$, and we let
$$
\begin{equation*}
\widehat{f}_{\mathbf{k}}:=\dfrac{1}{(2\pi)^{d}}\displaystyle\int_{\mathbb{T}^{d}} f(\mathbf{t})e^{-i\mathbf{kt}}\,\mathrm{d}\mathbf{t}
\end{equation*}
\notag
$$
denote the Fourier coefficients of the function $f\in L^{1}(\mathbb{T}^{d})$ with respect to the $d$-dimensional trigonometric system $\{e^{i\mathbf{kx}}\}_{\mathbf{k}=-\infty}^{\infty}$, where $\mathbf{kx}=k_{1}x_{1}+\cdots+k_{d}x_{d}$. Rectangular and spherical partial sums of the Fourier series of a function $f\in L^{1}\mathbb{(T}^{d}{)}$ with respect to the $d$-dimensional trigonometric system are defined by
$$
\begin{equation*}
S_{\mathbf{n}}(f;\mathbf{x})=\sum_{|\mathbf{k}|\leqslant\mathbf{n}}{\widehat{f}}_{\mathbf{k}}e^{i\mathbf{kx}},
\end{equation*}
\notag
$$
and, respectively, by
$$
\begin{equation*}
S_{(\lambda)}(f;\mathbf{x})=\sum_{k_{1}^{2}+\cdots+k_{d}^{2}\leqslant\lambda^{2}}{\widehat{f}}_{\mathbf{k}}e^{i\mathbf{kx}}.
\end{equation*}
\notag
$$
Pringsheim convergence (over rectangles) or convergence over spheres (in one sense or another) of the Fourier series of $f$ means that
$$
\begin{equation*}
S_{\mathbf{n}}(f;\mathbf{x})\to f\quad\text{as}\quad n_{\ast}=\min(n_{1},\dots,n_{d})\to\infty
\end{equation*}
\notag
$$
or, respectively,
$$
\begin{equation*}
S_{(\lambda)}(f;\mathbf{x})\to f\quad\text{as}\quad \lambda\to\infty.
\end{equation*}
\notag
$$
Set $\rho(\Omega)_{A}:=\limsup_{n_{\ast}\to\infty} \#(\Omega\cap\lbrack\mathbf{-n,n})^{d})/ \#(A\cap\lbrack\mathbf{-n,n})^{d})$, $D_{\langle A\rangle}=\sum_{\mathbf{k}\in A}e^{i\mathbf{kx}}$, and $S_{\langle A\rangle}(f;\mathbf{x})= \sum_{\mathbf{k}\in A}{\widehat{f}}_{\mathbf{k}}e^{i\mathbf{kx}}$, where $A\subset \mathbb{Z}^{d}$ is a bounded set, $\Omega\subset A$, $\#(E)$ is the cardinality of the finite set $E$, and $[\mathbf{-n,n})^{d}=[-n_{1},n_{1})\times\cdots\times\lbrack-n_{d},n_{d})$. We call the quantity $\rho(\Omega)_{A}$ the density of the subset $\Omega$ with respect to the set $A$. Let $\Lambda(f):=\{\mathbf{k}\in \mathbb{Z}^{d} \colon\widehat{f}_{\mathbf{k}}\ne 0\}$ be the spectrum of $f\in L^{1}(\mathbb{T}^{d})$. We say that $f\in L^{1}(\mathbb{T}^{d})$ is almost universal over rectangles (over spheres) for the class $L^{p}(\mathbb{T}^{d})$ with respect to the $d$-dimensional trigonometric system $\{e^{i\mathbf{kx}},\mathbf k\in\mathbb{Z}^{d}\}$ if there exists a sequence of signs $\{\delta_{\mathbf{k}}=\pm1;\mathbf k\in \mathbb{Z}^{d}\}$ such that $\rho(\Omega(f))_{\Lambda(f)}=1$, where $\Omega(f)=\{\mathbf{k}\in \Lambda(f)\colon \delta_{\mathbf{k}}=1\}$ and such that the rectangular (respectively, spherical) partial sums of the series $\sum_{\mathbf{k}\in \mathbb{Z}^{d}}\delta_{\mathbf{k}}\widehat{f}_{\mathbf{k}}e^{i\mathbf{kx}}$ are dense in $L^{p}(\mathbb{T}^{d})$. It follows from Kolmogorov’s famous theorem [6] that there exists no integrable function such that its Fourier series in the trigonometric system is universal in the class of all measurable functions. Getsadze [7] (also see Konyagin [1]) proved that an analogue of Kolmogorov’s theorem in [6] does not hold for the multiple trigonometric system. Nevertheless, the following result holds (see [2]). Theorem 1 (Konyagin). There exists no function $U\in L^{1}(\mathbb{T}^{d})$, $d>1$, whose Fourier series in the $d$-dimensional trigonometric system is universal over rectangles in $L^{p}(\mathbb{T}^{d})$ for $p\in(0,1)$. This is a consequence of the following stronger result. Theorem 2 (Konyagin [2]). Assume that a subsequence of Pringsheim partial sums of a function $f\in L^{1}(\mathbb{T}^{d})$ converges to a finite function $g$ on a set $E\subset \mathbb{T}^{d}$ of positive measure. Then $g=f$ almost everywhere on $E$. The following results also hold. Theorem 3 (Konyagin). For each function $f\in L^{1}(\mathbb{T}^{d})$, $d \geqslant2$, there exists a subsequence of Pingsheim partial sums that converges to it in $L^{p}(\mathbb{T}^{d})$ for each $p\in(0,1)$. Theorem 4 (Konyagin). Let $\{A_{k}\}_{k=1}^{\infty}$ be a sequence of bounded subsets of the space $\mathbb{R}^{d}$, let $N_{k}=\#(\mathbb{Z}^{d}\cap A_{k}) \geqslant3$, and assume that $\lim_{k\to\infty}\|D_{\langle A_{k}\rangle}\|_{1}/\log N_{k}=\infty$. Then there exist an increasing sequence $\{k_{j}\}_{j=1}^{\infty}$ and $f\in L^1(\mathbb{T}^{d})$ such that $\operatorname{lim}_{j\to\infty}|S_{\langle A_{k_{j}}\rangle} (f;\mathbf{x})|=\infty$ almost everywhere. The following results are consequences of Theorem 4. Corollary 1 (Konyagin). Let $d \geqslant2$, and let $\{\mathbf{n}^{(\upsilon)}\}_{\upsilon\in\mathbb{N}}\in \mathbb{Z}_{+}^{d}$ be sequence satisfying the condition $\min(n_{1}^{(\upsilon)},\dots,n_{d}^{(\upsilon)})\to\infty$ as $\upsilon\to\infty$. Then this sequence contains a subsequence $\{\mathbf{\check{n}}^{(\upsilon)}\}_{\upsilon\in\mathbb{N}}$ such that for some function $f$ the condition $\lim_{\upsilon\to\infty}|S_{\check{\mathbf{n}}^{(\upsilon)}} (f;\mathbf{x})|= \infty$ holds almost everywhere. Corollary 2 (Konyagin). Let $d\geqslant2$, and let $A\subset \mathbb{R}^{d}$ be a bounded body with non-empty interior. Then there exists a sequence $\{m_{j}\}_{j\in\mathbb{N}}$ such that for some function $f\in L^{1}(\mathbb{T}^{d})$ the condition $\lim_{j\to\infty}\!|S_{\langle m_{j}A\rangle}(f;\mathbf{x})|= \infty$ holds almost everywhere. In a similar way we can deduce from Theorem 4 that the absolute values of the partial sums over hyperbolic crosses diverge to infinity almost everywhere. Corollary 3 (Konyagin). Let $d \geqslant2$ и $A=\{\mathbf{u}=(u_{1},\dots,u_{d})\in \mathbb{R}^{d}\colon 0<|u_{1}\cdots u_{d}| \leqslant1\}\subset\mathbb{R}^{d}$. Then there exists a sequence $\{m_{j}\}_{j\in\mathbb{N}}$ such that for some function $f\in L^{1}\mathbb{(T}^{d})$ the condition $\lim_{j\to\infty}\!|S_{\langle m_{j}A\rangle}(f;\mathbf{x})|= \infty$ holds almost everywhere. By Theorem 1 there exists no universal function function with respect to the $d$-dimensional trigonometric function for the class $L^{p}(\mathbb{T}^{d})$, where $p\in(0,1)$. However, an almost universal function exists. More precisely, the following theorem holds (see [3]). Theorem 5 (Grigoryan). For any $p\in(0,1)$ and $d \geqslant2$ there exists a function $U\in L^{1}(\mathbb{T}^{d})$ that is almost universal over rectangles and spheres alike for the class $L^{p}(\mathbb{T}^{d})$, $p\in(0,1)$, with respect to the $d$-dimensional trigonometric system. It turns out that each measurable function which is finite almost everywhere can be transformed into an almost universal function by changing its values on a set of arbitrarily small measure. Namely, the following results hold. Theorem 6 (Grigoryan). For any $p\in(0,1)$, $d \geqslant2$, and $\delta>0$ and any measurable function $f$ on $\mathbb{T}^{d}$ which is finite almost everywhere, there exist a measurable function $g\in L^{1}(\mathbb{T}^{d})$ and a measurable set $E\subset\mathbb{T}^{d}$ of measure $|E|\geqslant(2\pi)^{d}-\delta$ such that $g=f$ on $E$ and $g$ is almost universal over rectangles and spheres alike for the class $L^{p}(\mathbb{T}^{d})$ for $p\in(0,1)$. Theorem 7 (Grigoryan). For any $p\in(0,1)$ and $d \geqslant2$ there exists a function $U\in L^{1}(\mathbb{T}^{d})$ that is almost universal over rectangles and spheres alike for the class of measurable functions on $\mathbb{T}^{d}$ with respect to the $d$-dimensional trigonometric system and which also has the following property: for any $\delta>0$ there exists a measurable set $E\subset \mathbb{T}^{d}$ of measure $|E|\geqslant(2\pi)^{d}-\delta$ such that for each function $f\in L^{1}(\mathbb{T}^{d})$ there exists a funtion $g\in L^{1}(\mathbb{T}^{d})$ such that $g=f$ on $E$ and the Fourier coefficients of $g$ and $U$ in the $d$-dimensional trigonometric system have the same moduli. 
Citation in format AMSBIB
\Bibitem{GriKon23} |
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