Multiple Trigonometric Fourier Series Research Articles

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Partial best approximations and the absolute Cesaro summability of multiple Fourier series

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).

Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

Almost Everywhere Convergence of Multiple Trigonometric Fourier Series of Functions from Sobolev Classes

Sufficient conditions for convergence of multiple Fourier series with Jk-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\subset M$ the Weyl multiplier for convergence of these series has the form $W(\nu)=\prod \limits_{j=1}^{N-k} \log(|\nu_{{\alpha}_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu=(\nu_1,\dots,\nu_N)\in{\mathbb Z}^N$. So, the "one-dimensional" Weyl multiplier -- $\log(|\cdot|+2)$ -- presents in $W(\nu)$ only on the places of "free" (nonlacunary) components of the vector $\nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more "free" components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

Convergence almost everywhere of multiple Fourier series over cubes

We study convergence almost everywhere of multiple trigonometric Fourier series over cubes defined on the d d -dimensional torus T d \mathbb {T}^d . We provide a new approach which allows us to prove the novel interpolation estimates for the Carleson maximal operators generated by the partial sums of the multiple Fourier series and all its conjugate series. Combining these estimates we show that these operators are bounded from a variant of the Arias-de-Reyna space Q A d Q\!A^d to the weak L 1 L^1 -space on T d \mathbb {T}^d . This implies that the multiple Fourier series of every function f ∈ Q A d f\in Q\!A^d and all its conjugate series converge over cubes almost everywhere. By a close analysis of the space Q A d Q\!A^d we prove that it contains a Lorentz space that strictly contains the Orlicz space L ( log L ) d log ⁡ log ⁡ log L ( T d ) L(\log \,L)^{d} \log \log \log \,L(\mathbb {T}^d) . This yields a significant improvement of a deep theorem proved by Antonov which was the best known result on the convergence of multiple Fourier series over cubes.

ON Λ-CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES

We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the \(\lambda \)-convergence for \(\lambda >1\). The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class \( L (\ln ^ + L) ^ d \ln ^ + \ln ^ + \ln ^ + L ([0,2 \pi)^d ) \) has been generalized to the case of the \( \Lambda \)-convergence for some sequences \(\Lambda\).

The multipliers of multiple trigonometric Fourier series

Abstract We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.

Sufficient Conditions for Convergence Almost Everywhere of Multiple Trigonometric Fourier Series with Lacunary Sequence of Partial Sums

Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in the classes $L_p$, $p>1$, are obtained in the case where rectangular partial sums $S_n(x;f)$ of this series have numbers $n=(n_1,\dots,n_N)\in \mathbb Z^N$, $N\geq 3$, such that of $N$ components only $k$ ($1\leq k\leq N-2$) are elements of some lacunary sequences. Earlier, in the case where $N-1$ components of the number $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained for functions in the classes $L_p$, $p>1$, by M. Kojima (1977), and for functions in Orlizc classes by D. K. Sanadze, Sh. V. Kheladze (1977) and N. Yu. Antonov (2014). Note that presence of two or more “free” components in the number $n$, as follows from the results by Ch. Fefferman (1971) and M. Kojima (1977), does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

General algorithm for the numerical integration of functions of several variables

An algorithm is proposed for the numerical integration of an arbitrary function represent-able as a sum of an absolutely converging multiple trigonometric Fourier series. The resulting quadrature formulas have identical weights, and the nodes form a Korobov grid that is completely defined by two positive integers, of which one is the number of nodes. In the case of classes of functions with dominant mixed smoothness, it is shown that the algorithm is almost optimal in the sense that the construction of a grid of N nodes requires far fewer elementary arithmetic operations than NlnlnN. Solutions of related problems are also given.

Equiconvergence of expansions in multiple trigonometric Fourier series and Fourier integral with "Jk-lacunary sequences of rectangular partial sums"

We give a review of recent results on equiconvergence of expansions in multiple trigonometric Fourier series and Fourier integral in the case of summation over rectangles.

Growth estimates for arbitrary sequences of multiple rectangular Fourier sums

Growth estimates are obtained on a set of full measure for arbitrary sequences of rectangular partial sums of multiple trigonometric Fourier series.

Localization for multiple Fourier series with “J k -lacunary sequence of partial sums” in Orlicz classes

We obtain structural and geometric characteristics of sets on which weak generalized localization almost everywhere is valid for multiple trigonometric Fourier series of functions in the classes \(L(\log ^ + L)^{3k + 2} (\mathbb{T}^N ),1 \leqslant k \leqslant N - 2,N \geqslant 3\), in the case where the rectangular partial sums of these series have an “index” in which exactly k components are elements of lacunary sequences.

Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

We study multiple trigonometric Fourier series of functions f in the classes \(L_p \left( {\mathbb{T}^N } \right)\), p > 1, which equal zero on some set \(\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0\) (µ is the Lebesgue measure), \(\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N\), N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series Sn(x; f) have index n = (n1, ..., nN) ∈ ℤN, in which k (k ≥ 1) components on the places {j1, ..., jk} = Jk ⊂ {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with Jk-lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” Jk) of lacunary sequences in the index n with the structural and geometric characteristics of the set \(\mathfrak{A}\), which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure \(\mathfrak{A}_1\) of the set \(\mathfrak{A}\).

Equiconvergence of expansions in multiple trigonometric Fourier series and integrals in the case of a lacunary sequence of partial sums

296 We investigate the equiconvergence onN = (-π, π) N of expansions in multiple trigonometric Fourier series and Fourier integrals of functions f ∈ Lp(� N ) and g ∈ L p (� N ), p > 1, N ≥ 2, g(x )= f (x) onN , in the case when the "partial sums" of these expansions, i.e., Sn(x; f ) and Jα(x; g), respectively, have "indices" n =( n1, n2, …, nN) ∈ � N and α = (α1, α2, …, αN) ∈ � N with components nj and αj satisfying the relation |α j - n j | ≤ C, j = 1, 2, …, N, where C is a constant inde� pendent of n or α. In particular, we consider the case when some of the components nj are elements of lacunary sequences.

Summability of multiple trigonometric Fourier series by linear methods

The paper deals with the problem of estimation of deviations of functions of several variables from linear means of their multiple trigonometric Fourier series. An approach of reducing this problem to the corresponding problem for functions of single variable is developed.

A note on estimates for the growth order of sequences of multiple rectangular Fourier sums

Let $$\left\{ {S_{n_k } \left( {f,x} \right)} \right\}_{k = 1}^\infty$$ be some sequence of rectangular partial sums of the multiple trigonometric Fourier series of a function f, and let {λ k } =1 ∞ be a nondecreasing sequence of positive numbers. We investigate conditions on a class φ(L) under which any function f from this class satisfies estimates of the form $$S_{n_k } \left( {f,x} \right) = o\left( {\lambda _k } \right)$$ a.e., where the right-hand side depends on k only.

On the Exponential Uniform Strong Summability of Multiple Trigonometric Fourier Series

Abstract In the paper, in particular, it is proved that the 𝑠-dimensional trigonometric Fourier series of the 2π-periodic continuous on [–π, π]𝑠 function 𝑓 is uniformly strong summable to the function 𝑓 exponentially in the power . Moreover, it is proved that this result is best possible.

A weak generalized localization of multiple Fourier series of continuous functions with a certain module of continuity

Let E be an arbitrary measurable set, E ⊂ T N = [−π, π)N, N ≥ 1, μE > 0, and let μ be a measure. In this paper, a weak generalized almost everywhere localization is studied, i.e., for given subsets E 1 ⊂ E, μE 1 > 0 we study the almost everywhere convergence of multiple trigonometric Fourier series of functions that are zero on E. We obtain sufficient conditions for the almost everywhere convergence of multiple Fourier series (summable over rectangles) of functions from {ie031-01}, as δ → 0 on E 1. These conditions are given in terms of the structure and geometry of the sets E 1 and E and are related to certain orthogonal projections of the sets; they are called the {ie031-02} property of the set E. Previously, one of the authors had introduced the {ie031-03}, k = 1, 2, properties of the set E, which are related to one-dimensional and two-dimensional projections of the sets E and E 1 respectively, as sufficient conditions for the almost everywhere convergence of Fourier series of functions from L 1(T N) and L p (T N), p > 1. The results presented generalize these ideas.

Convergence and localization of multiple Fourier series for classes of bounded Λ-variation

Previously obtained results for convergence and localization of multiple trigonometric Fourier series for functions from classes of bounded Λ-variation and embedding of these classes into each other are strengthened in the paper. The case when sequences Λ and M have a limit of the ratio Σ =1 1/λ n /Σ =1 1/µ n is considered. A more strict condition, the existence of a limit for the ratio λ n /µ n was considered before.