Equiconvergence Of Expansions Research Articles

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T N = [−π, π) N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L p (T N ) and g ∈ L p (R N ), p > 1, N ≥ 3, g(x) = f(x) on T N , in the case where the “partial sums” of these expansions, i.e., S n (x; f) and J α(x; g), respectively, have “numbers” n ∈ Z N and α ∈ R N (n j = [α j ], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N − 1 components which are elements of “lacunary sequences.”

Equiconvergence of expansions into triple trigonometric series and Fourier integral for continuous functions with a certain modulus of continuity

We study the problem of equiconvergence on {ie24-1} for expansions in a triple trigonometric Fourier series and a Fourier integral of continuous functions with a certain modulus of continuity in the case of a “lacunary sequence of partial sums”.

Equiconvergence of expansions in multiple trigonometric Fourier series and Fourier integral with "Jk-lacunary sequences of rectangular partial sums"

We give a review of recent results on equiconvergence of expansions in multiple trigonometric Fourier series and Fourier integral in the case of summation over rectangles.

Equiconvergence of expansions in multiple trigonometric Fourier series and integrals in the case of a lacunary sequence of partial sums

296 We investigate the equiconvergence onN = (-π, π) N of expansions in multiple trigonometric Fourier series and Fourier integrals of functions f ∈ Lp(� N ) and g ∈ L p (� N ), p > 1, N ≥ 2, g(x )= f (x) onN , in the case when the "partial sums" of these expansions, i.e., Sn(x; f ) and Jα(x; g), respectively, have "indices" n =( n1, n2, …, nN) ∈ � N and α = (α1, α2, …, αN) ∈ � N with components nj and αj satisfying the relation |α j - n j | ≤ C, j = 1, 2, …, N, where C is a constant inde� pendent of n or α. In particular, we consider the case when some of the components nj are elements of lacunary sequences.

Componentwise uniform equiconvergence of expansions in root vector functions of the Dirac operator with the trigonometric expansion

We consider the one-dimensional Dirac operator on a finite interval G = (a, b). We analyze the uniform componentwise equiconvergence of expansions in root vector functions of this operator with the trigonometric Fourier series on a compact set. Theorems on the componentwise equiconvergence on a compact set and the componentwise localization principle are proved.

The equiconvergence of expansions in eigenfunctions and associated functions of an integral operator with involution

In this paper we study an integral operator with involution. We solve the problem on the exact inversion of this operator, we obtain and study the integro-differential system for the Fredholm resolvent and, finally, we prove the theorem on the equiconvergence of expansions in eigenfunctions and associated functions, in the usual trigonometric system.

О равносходимости разложений для некоторого класса функционально-дифференциальных операторов с инволюцией на графе

О равносходимости разложений для некоторого класса функционально-дифференциальных операторов с инволюцией на графе

On equiconvergence of spectral expansions of self-adjoint integral operators

The paper gives simple sufficient conditions on the kernel of the self-adjoint integral operator considered ensuring the equiconvergence of expansions in eigenfunctions and associated functions with the ordinary trigonometric Fourier series expansions. The presented arguments correct inaccuracies in [1].

Equiconvergence of expansions in eigenfunctions of integral operators with kernels that can have discontinuities on the diagonals

Simple conditions are found ensuring the equiconvergence of the Fourier expansion of a function in in the eigenfunctions and the associated functions of an integral operator and the expansions of and in the standard trigonometric system.

Generalized localization and equiconvergence of expansions in double trigonometric series and in the Fourier integral for functions fromL(ln+ L)2

Generalized localization and equiconvergence of expansions in double trigonometric series and in the Fourier integral for functions fromL(ln+ L)2

Uniform equiconvergence of expansions with respect to a system of exponential functions and in the trigonometric fourier series

Uniform equiconvergence of expansions with respect to a system of exponential functions and in the trigonometric fourier series

On the equiconvergence of expansions by Riesz bases formed by eigenfunctions of a linear differential operator of order2n

The theory of non-selfadjoint differential operators has great importance in several applications. Developing a fruitful method of V. A. II'in [1], in [21 I. Jod described a new method which makes possible to avoid the use of the basic solution in such investigations. Using this method, a general equiconvergence theorem was proved in [9] for the SchrSdinger operator, having generalized the results of the papers [5], [6], [7] and [8]. This theorem concerns the expansions by Riesz bases formed by eigenfunctions of higher order of the SchrSdinger operator (but seems to be new also for the case of orthonormal bases). The existence of such Riesz bases was proved for a wide class of operators (also of higher order) by V. P. Mikhailov [10] and G. M. Keselman [111. Later on, the method of [2] was generalized for the case of differential operators of higher order in [41, [12] and [13]. These generalizations are based on a new formula which extends the well-known mean-value formula of E. C. Titchmarsh for more general operators. Using the method and the results of these papers, we shall prove a general equiconvergence theorem for linear differential operators of higher order. In this theorem the eigenvalues of the Riesz basis may be arbitrary complex numbers. Our result is new also for orthonormal bases.