On the kolmogorov theorems on Fourier series and conjugate functions

We consider Fourier series of summable functions from spaces ??wider?? than L ₁. We describe classes ??(L) which contain conjugate functions, where their conjugate Fourier series converge. The obtained results are more general than A. N. Kolmogorov theorems on the convergence of Fourier series in metrics weaker than that of L ₁.     

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 = {λ _n } are found for the convergence of multiple Fourier series of functions of bounded partial Λ-variation.     

Approximation of functions from L p (ω) β by matrix means of their Fourier series

We formulate some special conditions for the integrable functions and moduli of continuity. We give the results on rate of approximation of such functions by matrix means of their Fourier series, where the entries of the rows of the matrix generate the sequences belonging to the classes MRBVS and MHBVS, We also present some results on norm approximation for functions from the generalized integral Lipschitz classes.     

Summability of multiple Fourier series of functions of bounded generalized variation

The paper is devoted to summability problems for multiple Fourier series of functions of bounded generalized variation. We find necessary and sufficient conditions on a sequence Λ = {λ _n } that guarantee the convergence of the Cesàro means of multiple Fourier series of functions of bounded Λ-variation.     

Multidimensional pseudo-spectral methods on lattice grids

When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L²-integrable, the contributions from many of the Ns terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.     

Strong spherical means of multiple Fourier series

The present paper is a survey of some results related to the strong means of spherical partial sums, their integral analogs and the Bochner-Riesz sums of Fourier series of functions defined almost everywhere on the m-dimensional torus.     

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip ??. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.     

Summability of Fourier series for almost-periodic functions on locally compact Abelian groups

Some results concerning the summability of Fourier series of continuous 2π-periodic functions are generalized for the case of almost-periodic functions defined on locally compact Abelian groups.     

Strong summability and convergence of multiple trigonometric series over polyhedrons

The paper is a survey of the known results related with Fourier series strong polyhedral means of functions continuous on the m-dimensional torus, Sidon type inequalities for Dirichlet kernels in polyhedrons and integrability and convergence of multiple trigonometric series.     

О сходимости в метрике L четных тригонометрических интерполяционных полиномов по равноотстояшим уэлам

Even, 2π-periodic, continuous for all x ≠ 2πn, n = 0, 1, …, functions, represented by Fourier series are considered. The question of convergence in the metric L of the trigonometric interpolation cosine polinomials of such functions with convex, quasiconvex, monotone and quasimonotone Fourier coefficients is investigated.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Norms of zonal spherical functions and Fourier series on compact symmetric spaces

New estimates are given for the p-norms of zonal spherical functions on compact symmetric spaces. These estimates are applied to give a Cohen type inequality for convolutor norms which leads to negative results on p-mean convergence of Fourier series on compact symmetric spaces of arbitrary rank.     

On convergence of odd trigonometric interpolation polynomials with equidistant points in the metric <Emphasis Type="Italic">L</Emphasis>

Odd, 2π-periodical, continuous functions represented by Fourier series are considered. The question of the convergence in the metric L of trigonometric interpolation sine polinomials of functions with monotone and quasimonotone Fourier coefficients is considered.     

Continuous functions with universally divergent Fourier series on small subsets of the circle

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality properties.     

On Fourier series for higher order (partial) derivatives of functions

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.     

Lineability of Universal Divergence of Fourier Series

It is proved that large vector spaces of continuous complex functions on the unit circle exist such that all their nonzero members satisfy that, for many small subsets E of the circle, the partial sums of their Fourier series approximate any prescribed function on E. This completes or improves a number of results by several authors.     

On the divergence of Fourier series of functions in several variables

The paper deals with the problems of divergence of the series from absolute values of the Fourier coefficients of functions in several variables. It is proved that as the dimension of the space increases, the absolute convergence of Fourier series with respect to any complete orthnormal system (ONS) of functions with continuous partial derivatives becomes worse. For instance, for any ? ∈ (0, 2) there exists a function in variables $k > \frac{{2(2 - \varepsilon )}} {\varepsilon }$ having all the continuous partial derivatives, however the series of absolute values of its coefficients with respect to any complete orthnormal system diverges in power 2 ? ?.     

Application of Interpolational Methods to the Study of Properties of Functions of Several Variables

In this paper, interpolation theorems for spaces of functions of several variables are used to generalize and refine Hörmander's theorem on the multipliers of the Fourier transform from L _p to L _q and the Hardy--Littlewood--Paley inequality for a class of multiple Fourier series in the multidimensional case.     

A version of the Uncertainty Principle for functions with lacunary Fourier transforms

We prove estimates of the L² norms on relatively dense subsets of the real line of functions with Fourier transforms supported on lacunary sets of intervals. This result is a real line analogue of Zygmund's theorem on lacunary trigonometric series. The results also hold in higher dimensions.     

Degree of approximation of functions in the Hölder metric by (e,c) means

Degree of approximation of functions by the (e, c) means of its Fourier series in the Hölder metric is studied.