Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function e α · 2 defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for α < 1 / 2 . The result provides a wider choice of the constant α than that of Fernique.

Absolute convergence factors of Lipschitz class functions for general Fourier series

The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. It is established that the resulting conditions are best possible in certain sense.

Modeling the Quasigeoid Heights on Local Areas of the Earth Surface by the Results of Expansion into a Generalized Fourier Series

The article presents two methods of modeling discrete heights of a quasigeoid on a local area of the earth’s surface using a gen-eralized Fourier series. The first method is based on modeling the characteristics of the earth’s gravitational field on a plane and involves the use of a two-dimensional Fourier transform by an orthonormal system of trigonometric functions. The second method consists in the expansion of the quasigeoid heights in a Fourier series by an orthonormal system of spherical functions on a local area of the earth’s surface. The errors of approxima-tion of the obtained discrete values of the quasigeoid heights on the local territory are analyzed. It is shown that with the modern computing technology, the most accurate and technologically simple way to model the quasigeoid heights on local areas is to expand them into a Fourier series by an orthonormal system of spherical functions.

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

In this paper, we explore orthogonal systems in $$\mathrm {L}_2({\mathbb R})$$L2(R) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ and a sequence of polynomials $$\{p_n\}_{n\in {\mathbb Z}_+}$${pn}n∈Z+ orthonormal with respect to a symmetric probability measure $$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi $$dμ(ξ)=w(ξ)dξ. If $$\mathrm{d}\mu $$dμ is supported by the real line, this system is dense in $$\mathrm {L}_2({\mathbb R})$$L2(R); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from $$\mathrm{d}\mu $$dμ to $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.

Investigation of eigenvibrations of a loaded bar

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

Comment on “On the Frame Properties of Degenerate System of Sines”

The proof of Theorem 3.1 of the paper “On the Frame Properties of Degenerate System of Sines” (see (Bilalov and Guliyeva, 2012)) published earlier in this journal contains a gap; the reasoning given there to prove this theorem is not enough to state the validity of the mentioned theorem. To overcome this shortage we state the most general fact on the completeness of sine system which implies in particular the validity of this fact. It is shown in this note that the system{ω(t)φn(t)}, where{φn(t)}is an exponential or trigonometric (cosine or sine) systems, becomes complete in the corresponding Lebesgue spaceLp(-π,π)orLp(0,π), respectively, whenever{ω(t)φn(t)}belongs to the corresponding Lebesgue space for all indicesn(under the evident natural conditionmes⁡{t:ω(t)=0}=0). It is also shown that the same conclusion does not remain valid for, in general, any complete or complete orthonormal system{φn(t)}. Besides it, the largest class of functionsω(t)for which the system{ωtsin⁡nt}n∈Nis complete inLp(0,π)space is determined.