Convergence Of Multiple Fourier Series Research Articles

On the generalized β-absolute convergence of single and multiple Fourier series

In this paper, we provide sufficient conditions for the generalized β-absolute convergence of multiple Fourier series of a function of p-(Λ1, ... ,ΛN)-bounded variation.

Pick’s Theorem and Convergence of Multiple Fourier Series

We add another brick to the large building comprising proofs of Pick’s theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick’s theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

On the almost everywhere convergence of multiple Fourier series of square summable functions

On the almost everywhere convergence of multiple Fourier series of square summable functions

Some properties of an odd-dimensional space

Abstract In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2015, 9, 12–21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9–17]. The obtained results are the best possible in a certain sense.

The Convergence Problems of Eigenfunction Expansions of Elliptic Differential Operators

In the present research we investigate the problems concerning the almost everywhere convergence of multiple Fourier series summed over the elliptic levels in the classes of Liouville. The sufficient conditions for the almost everywhere convergence problems, which are most difficult problems in Harmonic analysis, are obtained. The methods of approximation by multiple Fourier series summed over elliptic curves are applied to obtain suitable estimations for the maximal operator of the spectral decompositions. Obtaining of such estimations involves very complicated calculations which depends on the functional structure of the classes of functions. The main idea on the proving the almost everywhere convergence of the eigenfunction expansions in the interpolation spaces is estimation of the maximal operator of the partial sums in the boundary classes and application of the interpolation Theorem of the family of linear operators. In the present work the maximal operator of the elliptic partial sums are estimated in the interpolation classes of Liouville and the almost everywhere convergence of the multiple Fourier series by elliptic summation methods are established. The considering multiple Fourier series as an eigenfunction expansions of the differential operators helps to translate the functional properties (for example smoothness) of the Liouville classes into Fourier coefficients of the functions which being expanded into such expansions. The sufficient conditions for convergence of the multiple Fourier series of functions from Liouville classes are obtained in terms of the smoothness and dimensions. Such results are highly effective in solving the boundary problems with periodic boundary conditions occurring in the spectral theory of differential operators. The investigations of multiple Fourier series in modern methods of harmonic analysis incorporates the wide use of methods from functional analysis, mathematical physics, modern operator theory and spectral decomposition. New method for the best approximation of the square-integrable function by multiple Fourier series summed over the elliptic levels are established. Using the best approximation, the Lebesgue constant corresponding to the elliptic partial sums is estimated. The latter is applied to obtain an estimation for the maximal operator in the classes of Liouville.

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 the absolute convergence of Fourier series with respect to general orthonormal systems

Abstract The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.

Absolute convergence of multiple Fourier series of functions of $$\phi \left( \varLambda ^1,\ldots ,\varLambda ^N\right) $$ ϕ Λ 1 , … , Λ N -bounded variation

In this paper, we provide a sufficient condition for the absolute convergence of multiple Fourier series of a function of $$\phi \left( \varLambda ^1,\ldots ,\varLambda ^N\right) $$ -bounded variation. This condition generalizes one for a single Fourier series proved by Schramm and Waterman. We apply the classical technique of summation by parts and some key inequalities for convex functions (including the Jensen’s inequality) and $$\varDelta _2$$ -functions to prove our main result.

On absolute convergence of multiple Fourier series

We consider absolute convergence of multiple series of Fourier-Haar coefficients for functions of many variables with partial derivatives of higher order. We show that the obtained results are best possible for general orthonormal systems.

Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation

The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.

On the convergence of multiple Fourier series of functions of bounded partial generalized variation

The convergence of multiple Fourier series of functions of bounded partial Λ-variation is investigated. The sufficient and necessary conditions on the sequence Λ = {λn} are found for the convergence of multiple Fourier series of functions of bounded partial Λ-variation.

On necessary conditions for the convergence of Fourier series

We establish necessary conditions for the convergence of multiple Fourier series of integrable functions in the mean.

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.

Absolute convergence of multiple Fourier series revisited

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szasz, Zygmund and others related to absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.

On the absolute convergence of multiple fourier series

Let f: RN → C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.

Continuity in -variation of functions of several variables and convergence of multiple Fourier series

The behaviour of rectangular partial sums of the Fourier series of functions of several variables having bounded -variation is considered. It is proved that if a continuous function is also continuous in harmonic variation, then its Fourier series uniformly converges in the sense of Pringsheim. On the other hand, it is demonstrated that in dimensions greater than 2 there always exists a continuous function of bounded harmonic variation with Fourier series divergent over cubes at the origin.

Convergence of multiple Fourier series: main results and unsolved problems

Convergence of multiple Fourier series: main results and unsolved problems

On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy.