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This article, published in Mathematical Notes, discusses the absolute convergence of Fourier series of functions of two variables in the space C1, ¿. The authors explore the conditions that ensure the absolute convergence of the series and define the space A, which consists of functions for which the series converges. They also discuss the classical results of Bochner and Weinger, who extended the Bernstein theorems to functions of two variables. The authors refine Weinger's result using probabilistic considerations and provide a theorem that demonstrates the existence of a function in the space C1, ¿ that is not included in A. The article also mentions the sufficient conditions on the modulus of continuity ¿(¿) that ensure the embedding C1, ¿ ¿ A. The work was financially supported by the Russian Science Foundation. [Extracted from the article]

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened. [ABSTRACT FROM AUTHOR]

In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier-Laplace sums in the spaces S(σ), m ≥ 3, in terms of best approximations and moduli of continuity and consider the constructive characteristics of function classes defined by the moduli of continuity of their elements. The given statements generalize the results of the author's work carried out in 2007. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered. [ABSTRACT FROM AUTHOR]

We obtain sharp Jackson-Stechkin type inequalities for moduli of continuity of kth order Ω in which, instead of the shift operator T f, the Steklov operator S( f) is used. Similar smoothness characteristic of functions were studied earlier in papers of Abilov, Abilova, Kokilashvili, and others. For classes of functions defined by these characteristics, we calculate the exact values of certain n-widths. [ABSTRACT FROM AUTHOR]

It is well known that the Fourier coefficients of continuous functions with respect to classical orthogonal systems (trigonometric, Haar, and Walsh) can be estimated via the moduli of continuity of the functions. However, not all orthonormal systems possess this property. We obtain necessary and sufficient conditions on orthonormal systems such that the Fourier coefficients of continuous functions with respect to these orthonormal systems can be estimated via the moduli of continuity in a certain sense. [ABSTRACT FROM AUTHOR]

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of mth order in the space L as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L. [ABSTRACT FROM AUTHOR]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L2[0, 2π] satisfying the constraint , where 0 < h < ∞, 1/ r < p ≤ 2, r ∈ ℕ, and ω m( f( r); t) is the modulus of continuity of mth order of the derivative f( r)( x) ∈ L2[0, 2 π]. [ABSTRACT FROM AUTHOR]

We obtain order-sharp estimates of best approximations for the classes B of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω( t). [ABSTRACT FROM AUTHOR]

We obtain sharp inequalities of Jackson-Stechkin type in which the modulus of continuity of the function is defined in terms of the Steklov function. For classes of functions given by this characteristic, we obtain sharp values of the n-widths. [ABSTRACT FROM AUTHOR]

For a gap sequence of natural numbers { n k}, for a nondecreasing function φ: [0,+∞) → [0,+∞) such that φ( u) = o( u ln ln u) as u → ∞, and a modulus of continuity satisfying the condition (ln k)−1 = O( ω( n)), we present an example of a function F ∈ φ( L) ∩ H with an almost everywhere divergent subsequence { S n k( F, x)} of the sequence of partial sums of the trigonometric Fourier series of the function F. [ABSTRACT FROM AUTHOR]

Suppose that H p ( E ) is the Hardy space for the first octant and P ( f, x), l > 0, is the generalized Abel-Poisson means of a function f ∊ H p ( E ). In this paper, we prove the inequalities where $$\widetilde\omega _l (\varepsilon ,f)_p $$ and ω l ( ɛ, f) p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik. [ABSTRACT FROM AUTHOR]

An analog of Szasz’s theorem on the absolute convergence of trigonometric Fourier series is established for expansions in the eigen and associated functions of integral operators some of whose kernels involve derivatives with discontinuities on the diagonals. [ABSTRACT FROM AUTHOR]

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

In this paper we study functions belonging to the classes Vε and Λ BV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classes Hω in the classes Vϕ and ABV are obtained. [ABSTRACT FROM AUTHOR]

The article deals with the Jackson theorem in the space L2 on the interval [-1,1] with power-law weight. It provides equations for Bessel function and Hilbert space. Further, this analysis is intended to compare inequality with the well-known exact Jackson inequality in the space L2 of periodic functions, which was proved by Cherhykh.