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In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL. [ABSTRACT FROM AUTHOR]

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given. [ABSTRACT FROM AUTHOR]

We consider the problem of the mean-square approximation of complex functions regular in a domain D ⊂ C by Fourier serieswith respect to an orthogonal (inD) systemof functions {ϕk(z)}, k = 0, 1, 2,.... For the case inwhich D = {z ∈ C: |z| < 1}, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system {zk}, k = 0, 1, 2,..., for classes of functions defined by a special mth-order modulus of continuity. Exact values of the series of n-widths for these classes of functions are calculated. [ABSTRACT FROM AUTHOR]

We present a necessary and sufficient condition for the series of absolute values of blocks of Fourier series elements and blocks of series of summands in Parseval’s identity to converge in the class of two-variable functions of bounded variation in the sense of Hardy. [ABSTRACT FROM AUTHOR]

In this paper, we study various generalizations of the classical Wiener algebra on a Banach space and prove analogs of Wiener's theorem on the invertibility of elements of such algebras. [ABSTRACT FROM AUTHOR]

An unbounded operator is said to be bisectorial if its spectrum is contained in two sectors lying, respectively, in the left and right half-planes and the resolvent decreases at infinity as 1/ λ. It is known that, for any bounded function f, the equation u′ − Au = f with bisectorial operator A has a unique bounded solution u, which is the convolution of f with the Green function. An example of a bisectorial operator generating a Green function unbounded at zero is given. [ABSTRACT FROM AUTHOR]

For higher-order partial differential equations in two or three variables, the Dirichlet problemin rectangular domains is studied. Small denominators hampering the convergence of series appear in the process of constructing the solution of the problem by the spectral decomposition method. A uniqueness criterion for the solution is established. In the two-dimensional case, estimates justifying the existence of a solution of the Dirichlet problem are obtained. In the three-dimensional case where the domain is a cube, it is shown that the uniqueness of the solution of the Dirichlet problem is equivalent to the great Fermat problem. [ABSTRACT FROM AUTHOR]

The paper deals with the spherical Fourier sums $$S_r (f,x) = \sum\nolimits_{\left\| k \right\| \leqslant r} {\hat f} (k)e^{ik \cdot x}$$ of a periodic function f in m variables and the strong integral means of these sums ((∫ | S( f, x)| dr)/ R for p ≥ 1. We establish the exact growth order as R → +∞ of the corresponding operators, i.e., the growth order of the quantities sup((∫| S( f, 0)| dr)/ R). The upper and lower bounds differ by their coefficients, which depend only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given. [ABSTRACT FROM AUTHOR]

For the two-dimensional nonstationary Euler equations of an ideal incompressible fluid, the flow problem with a given vortex is considered. The uniqueness of the solution is proved. [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 establish a nontrivial estimate for a short trigonometric sum of the form Ω e( α[ n]), where u ≥ √2 cx ℒ, A ≥ 1 is a fixed number, ℒ = ln x and c is a noninteger satisfying the conditions [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for β-absolute convergence (0 < β ≤ 1) of multiple Fourier series of functions of the classes . [ABSTRACT FROM AUTHOR]

We construct an example of a function from the class $Lo(\sqrt {\ln ^ + L} )$ whose Fourier-Vilenkin series diverges almost everywhere. [ABSTRACT FROM AUTHOR]

The article examines the coefficients of the expansion of smooth functions of several variables in the bases of the spaces Lp([0,1]d). It discusses the normalized basis in Lp([0,1]d) and the series of Fourier coefficients in the trigonometric system. It also discusses the Faber-Schauder system, Haar system, and Rademacher function.

For an equation of mixed type, namely, in the domain {( x, t) | 0 < x < 1, − α < t < β}, where α, β are given positive real numbers, we study the problem with boundary conditions . We establish a uniqueness criterion for the solution constructed as the sum of Fourier series. We establish the stability of the solution with respect to its nonlocal condition φ( x). [ABSTRACT FROM AUTHOR]

To any complex function there corresponds a Fourier series, which is often associated with a sequence { T} of Toeplitz n × n matrices. Functions whose Fourier series generate sequences of normal Toeplitz matrices are classified, and a procedure for constructing Fourier series for which the sequence { T} contains an infinite subsequence of normal matrices is described. [ABSTRACT FROM AUTHOR]

We establish a lower bound for the uniform norm of the trigonometric polynomial of special form via the sum of the L-norms of its summands. This result generalizes a theorem due to Kashin and Temlyakov, which, in turn, generalizes the classical Sidon inequality. [ABSTRACT FROM AUTHOR]

We obtain a new class of sets of uniqueness for double trigonometric series in the case of rectangular convergence as well as prove the two-dimensional analog of Privalov’s theorem. [ABSTRACT FROM AUTHOR]

The paper deals with the Sturm-Liouville operator generated in the space L2 = L2[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev space W [0, 1] for some integer p ≥ 0 and satisfy the conditions q( k)(0) = q( k)(1) = 0 for 0 ≤ k ≤ s − 1, where s ≤ p. Let the functions Q and S be defined by the equalities and let q n, Q n, and S n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system $$ \{ e^{2\pi inx} \} _{ - \infty }^\infty $$. Assume that the sequence q2 n − S2 n + 2 Q0 Q2 n decreases not faster than the powers n−s−2. Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L2[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition holds. [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]

We obtain upper bounds for character sums modulo a composite number over sets of numbers with missing digits in a number system. We derive results on the solvability of congruences of the form x 1 ⋯ x t ≡ λ (mod m) in the numbers with missing digits and also asymptotic formulas for the number of solutions. [ABSTRACT FROM AUTHOR]

For a function ƒ ∈, L 1( $$\mathbb{T}$$ ), we investigate the sequence ( C, 1) of mean values Φ(¦ S k ( x, ƒ) − ƒ( x)¦), where Φ( t): [0, +∞) → [0,+∞), Φ(0) = 0, is a continuous increasing function. We prove that if Φ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier. [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 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.