Eurasian Mathematical Journal

Invariant Subspaces in Non-Quasianalytic Spaces of \( \Omega \)-Ultradifferentiable Functions on an Interval

Natalia Abuzyarova, Ziganur Fazullin — Sun, 17 Nov 2024 00:00:00 +0000

We consider and solve a weakened version of the classical spectral synthesis problem for differentiation operator in non-quasianalytic spaces of ultradifferentiable functions (UDF). Moreover, we deal with the widest class of UDF among all known ones. Namely, we study the spaces of \( \Omega \)-ultradifferentiable functions introduced by Alexander Abanin in 2007–08. For subspaces of these spaces which are invariant under the differentiation operator we establish general conditions of weak spectral synthesis.

Optimal cubature formulas for Morrey type function classes on multidimensional torus

Sholpan Balgimbayeva, Daurenbek Bazarkhanov — Sun, 17 Nov 2024 00:00:00 +0000

In the paper, we establish estimates, sharp in order, for the error of optimal cubature formulas for the smoothness spaces \( B^s_{p,q,\tau}(\mathbb{T}^m) \) of Nikol’skii–Besov type and \( F^s_{p,q}(\mathbb{T}^m) \) of Lizorkin–Triebel type, both related to Morrey spaces, on multidimensional torus, for some range of the parameters \( s, p, q, \tau \) \( (0 < s < \infty, 1 \leq p, q \leq \infty, 0 \leq \tau \leq 1/p) \). In particular, we obtain those estimates for the isotropic Lizorkin–Triebel function spaces \( F^s_{\infty, q}(\mathbb{T}^m) \).

Weak version of symmetric space

Turdebek Bekjan — Sun, 17 Nov 2024 00:00:00 +0000

In this paper, we de ned weak versions of symmetric spaces and established Hölder and Chebyshev type inequalities for noncommutative spaces associated with these spaces.

Symmetry groups of pfaffians of symmetric matrices

Askar Dzhumadil'daev — Sun, 17 Nov 2024 00:00:00 +0000

We prove that the symmetry group of the pfaffian polynomial of a symmetric matrix is a dihedral group

The spectrum and principal functions of a nonself-adjoint Sturm–Liouville operator with discontinuity conditions

Nida Palamut Kosar, Ozge Akcay — Sun, 17 Nov 2024 00:00:00 +0000

This paper deals with the nonself-adjoint Sturm–Liouville operator (or one-dimensional time-independent Schrödinger operator) with discontinuity conditions on the positive half line. In this study, the spectral singularities and the eigenvalues are investigated and it is proved that this problem has a nite number of spectral singularities and eigenvalues with nite multiplicities under two additional conditions. Moreover, we determine the principal functions with respect to the eigenvalues and the spectral singularities of this operator.

Barrier composed of perforated resonators and boundary conditions

Igor Popov, Ekaterina Trifanova, Alexander Bagmutov, Irina Blinova — Sun, 17 Nov 2024 00:00:00 +0000

We consider the Laplace operator with the Neumann boundary condition in a two-dimensional domain divided by a barrier composed of many small Helmholtz resonators coupled with both parts of the domain through small windows of diameter \( 2a \). The main terms of the asymptotic expansions in \( a \) of the eigenvalues and eigenfunctions are considered in the case in which the number of the Helmholtz resonators tends to infinity. It is shown that such a homogenization procedure leads to some energy-dependent boundary condition in the limit. We use the method of matching the asymptotic expansions of boundary value problem solutions.

Weaving frames linked with fractal convolutions

Geetika Verma, Andrew Eberhard — Sun, 17 Nov 2024 00:00:00 +0000

Weaving frames have been introduced to deal with some problems in signal processing and wireless sensor networks. More recently, the notion of fractal operator and fractal convolutions have been linked with perturbation theory of Schauder bases and frames. However, the existing literature has established limited connections between the theory of fractals and frame expansions. In this paper, we define weaving frames generated via fractal operators combined with fractal convolutions. The aim is to demonstrate how partial fractal convolutions are associated to Riesz bases, frames, and the concept of weaving frames in a Hilbert space. The context deals with one-sided convolutions, i.e., both left and right partial fractal convolution operators on Lebesgue space \( L^p \) \( (1 \leq p \leq \infty) \). Some applications using partial fractal convolutions with null function have been obtained for the perturbation theory of bases and weaving frames.

Kolmogorov widths of anisotropic function classes and finite-dimensional balls

Anastasia Vasil’eva — Sun, 17 Nov 2024 00:00:00 +0000

In this paper, we obtain order estimates for the Kolmogorov widths of anisotropic periodic Sobolev and Nikol'skii classes, as well asanisotropic finite-dimensional balls.