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The article deals with the problem of the deviation from a function belonging to the space of partial sums of Fourier series with respect to the system of polynomials , orthogonal with respect to an inner product of Sobolev type. Here for and where is the Jacobi polynomial of degree . The main attention is paid to obtaining an upper bound for a Lebesgue-type function of partial sums of the Fourier series with respect to the system . [ABSTRACT FROM AUTHOR]

In this paper, we study the almost everywhere convergence of spherical partial sums of multiple Fourier series of functions from Sobolev classes. It is proved that almost everywhere convergence will take place under the same conditions on the order of smoothness of the expanded function as for multiple Fourier integrals; these conditions were found in a well-known paper of Carbery and Soria (1988). Our reasoning is largely based on the methodology developed in the work of Kenig and Thomas (1980). [ABSTRACT FROM AUTHOR]

Approximations on the closed interval of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression for the majorant of uniform approximations and an optimal parameter value ensuring the greatest rate of approximation by the method used in the paper are found. In the case of the even multiplicity of the poles of the approximating function, the asymptotic estimate is sharp. Examples of approximations of concrete functions are given. [ABSTRACT FROM AUTHOR]

A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums. [ABSTRACT FROM AUTHOR]

In this paper, two known theorems on |N̄, pn|k summability methods of Fourier series have been generalized for |A, pn|k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods. [ABSTRACT FROM AUTHOR]

It is established that the formulas determining the jump of a periodic function from the derivatives of the partial sums of its Fourier series and valid for functions of harmonic bounded variation (the HBV class) possibly will not hold for functions of Φ-bounded variation (in the sense of Schramm) if this class is wider than the HBV class. [ABSTRACT FROM AUTHOR]

We investigate the equiconvergence on T = [−π, π) of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L(T) and g ∈ L(R), p > 1, N ≥ 3, g( x) = f( x) on T, in the case where the 'partial sums' of these expansions, i.e., S( x; f) and J( x; g), respectively, have 'numbers' n ∈ Z and α ∈ R ( n = [α], j = 1,..., N, [ t] is the integral part of t ∈ R) containing N − 1 components which are elements of 'lacunary sequences.' [ABSTRACT FROM AUTHOR]

F. Schipp in 1969 proved the almost everywhere p-strong summability of Walsh-Fourier series and showed that if λ( n) →∞, then there exists a function f ∈ L[0, 1) for which the Walsh partial sums S( x, f) satisfy the divergence condition almost everywhere on [0, 1). In the present paper, we show that this condition holds everywhere. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The article studies the behavior of the rectangular partial sums of multiple Fourier series with respect to the system on the set A in the case where some of the components are elements of lacunary sequences. It focuses on the subsequences of partial sums of Fourier series which has better properties of convergence compared to the sequence of partial sums. Also for a single Walsh-Fourier series, it finds that for any sequence of Lebesgue constants, the sequence of partial sums converges.

Under minimal constraints on the coefficients, we prove uniqueness theorems for multiple trigonometric series in which, instead of pointwise convergence, we consider the convergence of integral means of spherical, cubic, and other partial sums. We also obtain sufficient conditions for the integrability of multiple trigonometric series, i.e., conditions under which these series are Fourier series. [ABSTRACT FROM AUTHOR]

The article presents the structure and geometry of maximal sets of convergence and unbounded divergence of multiple Fourier series with a J κ-lacunary sequence of rectangular partial sums. It notes that the criterion for the weak localization of the multiple Fourier series resulted in a theorem which allows the descriptions of maximal sets.

It has been established that, on the subsets $$ \mathbb{T}^N = [ - \pi ,\pi ]^N $$ describing a cross W composed of N-dimensional blocks, $$ W_{x_s x_t } = \Omega _{x_s x_t } \times [ - \pi ,\pi ]^{N - 2} (\Omega _{x_s x_t } $$ is an open subset of [− π, π]2) in the classes $$ L_p (\mathbb{T}^N ),p > 1 $$, a weak generalized localization holds, for N ≥ 3, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums $$ S_n (x;f)(x \in \mathbb{T}^N ,f \in L_p ) $$ of these series corresponds the number n = ( n1,..., n N) ∈ ℤ, some components n j of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating W) of the N-dimensional blocks $$ W_{x_s x_t } $$ as well as of the structure and geometry of $$ W_{x_s x_t } $$. In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the structural and geometric characteristics of some subsets of $$ \mathbb{T} $$ N = [ −π, π] N (of positive measure), on which, for the classes L p ( $$ \mathbb{T} $$ N ), p > 1, where N ≥ 3, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums S n ( x; f) ( x ∈ $$ \mathbb{T} $$ N , f ∈ L p ) of these series have a “number” n = ( n 1,..., n N ) ∈; ℤ such that some components n j are elements of lacunary sequences. For N = 3, similar studies are carried out for generalized localization almost everywhere. [ABSTRACT FROM AUTHOR]

We construct an example of a function from the class $H\_1^\{\backslash omega^*\}$, where $\backslash omega^*(t)=\backslash sqrt\{\backslash log\backslash log(t^\{-1\})/\backslash log(t^\{-1\})\}$,

The article focuses on the deviation of partial sums of Fourier-Haar systems with arbitrary dilation factor. It states that Fourier series converges uniformly for any continuous 1-periodic function. It adds that at the points of discontinuity, the values of the functions are taken as the half-sum of the left and right limits.