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We obtain an estimate for the convergence rate of the Fourier series of a continuous periodic function in terms of the modulus of continuity of the function and the value of its -variation. We prove that the leading term of the estimate is sharp. [ABSTRACT FROM AUTHOR]

Let be a smooth bounded domain in , and let be a linear elliptic differential operator of order with singular coefficients acting in . Under some assumptions of singularity for the coefficients of , we obtain Green's function estimates that hold up to the boundary of the domain and study the absolute convergence of the corresponding Fourier series. [ABSTRACT FROM AUTHOR]

We study how the well-known criterion for the uniform convergence of a sine series with monotone coefficients changes if, instead of monotonicity, one imposes the condition of -monotonicity with . Moreover, we obtain an addition to the well-known Kolmogorov theorem on the integrability of the sum of a cosine series with convex coefficients tending to zero. [ABSTRACT FROM AUTHOR]

An estimate for the convergence rate of the Fourier series of a continuous periodic function of bounded variation is refined. [ABSTRACT FROM AUTHOR]

A problem for an equation of high even order involving a Caputo fractional derivative in a rectangular domain with conjugation conditions is studied. A criterion for the uniqueness of the solution is given. The solution is constructed in the form of a Fourier series in eigenfunctions of a one-dimensional problem. [ABSTRACT FROM AUTHOR]

For a class of character systems of compact Abelian groups and for homogeneous Banach spaces satisfying some additional regularity conditions, we prove the following alternative: either the Fourier series of an arbitrary function in converges almost everywhere, or there exists a function in whose Fourier series diverges everywhere. We also prove that the classes of divergence sets of Fourier series in such function systems in the above-mentioned spaces are closed under at most countable unions and contain all sets of measure zero. As corollaries, we obtain some well-known and new results on everywhere divergent Fourier series in the trigonometric system as well as in the Walsh and Vilenkin systems and their rearrangements. [ABSTRACT FROM AUTHOR]

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]

General Rogosinsky–Bernstein linear polynomial means of Fourier series are introduced and three convergence criteria as are obtained: for convergence in the space of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and -points). For smooth functions, the rate of convergence in norm of , as well as of their interpolation analogues, is also studied. For approximation of functions in , the asymptotics is found along with the rate of decrease of the remainder term. [ABSTRACT FROM AUTHOR]

A Dirichlet-type problem for an equation of high even order with discontinuous coefficients is studied. A criterion for the uniqueness of the solution is given. The solution in the form of the Fourier series in the eigenfunctions of the one-dimensional problem is constructed. The problem of small denominators arises when justifying the convergence of the series. Sufficient conditions for the denominator to be distinct from zero are obtained. It is shown that the solvability of the problem is influenced not only by the dimension of the rectangle, but also by the orders of the given derivatives at the lower boundary of the rectangle. [ABSTRACT FROM AUTHOR]

The paper examines the following question: Under what orders of monotonicity are the upper and lower bounds of the sum of a cosine series near zero valid if they are obtained using the function ? [ABSTRACT FROM AUTHOR]

The question of the representability of a continuous function on in the form of the Fourier integral of a finite Borel complex-valued measure on is reduced in this article to the same question for a simple function. This simple function is determined by the values of the given function on the integer lattice . For , this result is already known: it is an inscribed polygonal line. The article also describes applications of the obtained theorems to multiple trigonometric Fourier series. [ABSTRACT FROM AUTHOR]

A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class of multipliers of Fourier–Haar series are obtained. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for a formal series with respect to the system of irreducible representations of a compact zero-dimensional group to be the Fourier–Stieltjes series of an additive measure is found. It is shown that, in the case of pointwise convergence of such a series everywhere on the group, its sum is integrable in the sense of Henstock-type integral, and the given series is the Fourier–Henstock series of its sum. [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]

Conditions (in particular, on the order of decrease of the coefficients) for the uniform Pringsheim convergence of double trigonometric series with rarely changing coefficients are found. [ABSTRACT FROM AUTHOR]

In a simply connected bounded domain with rectifiable Jordan boundary , we study the classes , , consisting of analytic functions in each of which, for any , satisfies the condition . Here is the generalized modulus of continuity of th order in and is a majorant. For these classes, we find upper and lower bounds for various -widths, as well as upper bounds for the moduli of Fourier coefficients. We obtain a constraint on the majorant under which the exact values of these extremal characteristics can be calculated. In the case of the unit disk, similar results are obtained for classes of analytic functions whose definitions include the Hadamard compositions in addition to and . Concrete realizations of some obtained exact results are presented. [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]

The class of generalized bounded variation is considered. For functions from this class, the deviation of the generalized Cesáro means of negative order in the norm of Lr, (1 ≤ r ≤ ∞) is estimated. [ABSTRACT FROM AUTHOR]

Estimates of the norms and the best approximations of the generalized Liouville–Weyl derivative in the two-dimensional case are obtained. [ABSTRACT FROM AUTHOR]

The zeta-function ζ(s, F), s = σ + it of a cusp form F of weight κ in the half-plane σ > (κ + 1)/2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(ζ(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials mn,Nα(x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ωδ = {0, δ, 2δ,...} with weight N (x) = e x (N x + + 1) (N x + 1) (1 e ) + 1 , where = 1 N , N 1. The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λn,Nα(x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θn/2, ∞) and θn = 4n + 2α + 2. [ABSTRACT FROM AUTHOR]

Properties of functions from the Sobolev orthogonal system W r generated by the Walsh system are studied. In particular, recurrence relations for functions from W 1 are obtained. The uniform convergence of Fourier series in the system W r to functions f from the S obolev spaces W L p r , p ≥ 1, r = 1, 2,... is proved. [ABSTRACT FROM AUTHOR]

The paper deals with sequences of positive numbers (dn) such that, multiplying the Fourier coefficients (Cn(f)) of functions from given function classes by these numbers, one obtains a convergent series of the form ∑ | C n (f) | p d n , 1 ≤ p < 2 . It is established that the resulting conditions cannot be strengthened in a certain sense. The results of the paper imply, in particular, some well-known results for trigonometric Fourier series. [ABSTRACT FROM AUTHOR]

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]

It is proved that, for each number p > 1, there exists a function L1[0, 1] whose Fourier-Walsh series is quasiuniversal with respect to subseries-signs in the class Lp[0, 1] in the sense of Lp-convergence. [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]

Conditions for the convergence of Fejér means for functions on the infinite-dimensional torus over cubes and rectangles are obtained, and a generalization of these results to the case of products of abstract measure spaces is proposed. [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, 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]

We study Fourier series of Jacobi polynomials P ( x), k = r, r +1,..., orthogonal with respect to the Sobolev-type inner product of the following form: $$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{\left( v \right)}}} \left( { - 1} \right){g^{\left( v \right)}}\left( { - 1} \right) + \int_{ - 1}^1 {{f^{\left( r \right)}}} {g^{\left( r \right)}}\left( t \right){\left( {1 - t} \right)^\alpha }dt$$ . It is shown that such series are a particular case of mixed series of Jacobi polynomials P ( x), k = 0, 1,..., considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials P ( x). [ABSTRACT FROM AUTHOR]

It is shown that the exact order of decrease of the norm in L of the remainder of a Fourier sine series with monotone coefficients can be expressed in terms of the coefficients of the series just as for a series with convex coefficients. But the numerical multipliers in the estimates are different. [ABSTRACT FROM AUTHOR]

In this paper, the following problem is studied. For what multipliers {λ} do the linear means of the Fourier series of functions f ∈ L [− π, π], , converge as n→∞ at all points at which the derivative of the function ∫ f exists? In the case λ = (1 − | k|/( n + 1)), a criterion of the convergence of the ( C, 1)-means and, in the general case λ = ϕ( k/( n + 1)), a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether ϕ( x) and xϕ′( x) belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given. [ABSTRACT FROM AUTHOR]

The article explores the conditions for integrability of a function on compact lie groups in terms of their Fourier coefficients. Topics mentioned include the inequalities that resulted from such conditions, formulation of the inequality's analog on compact Lie groups, and the Fourier coefficients of functions and distributions on the Lie group.

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than L = L[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ( L) of Kolmogorov's theorem on the convergence of Fourier series in L, p ∈ (0, 1), cannot be improved. [ABSTRACT FROM AUTHOR]

Generalized boundary conditions on multilayer films bounding a half-space and consisting of alternating infinitely thin strongly and weakly permeable layers are derived. The solution of the problem for the Laplace equation in a half-plane D bounded by a three-layer film is expressed in simple quadratures in terms of the solution of the classical Dirichlet problem in D without a film. [ABSTRACT FROM AUTHOR]

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]

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy-Littlewood theorem is not valid for p ≤ 2 m/( m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case. [ABSTRACT FROM AUTHOR]