Representation of the density functions of a multidimensional strictly stable distributions by series of generalized functions

В статье рассматриваются многомерные строго устойчивые распределения. Как известно, функции плотности этих законов не представляются в явном виде за исключением известных законов Гаусса и Коши. Отправным пунктом для исследований являются характеристические функции. Имеется несколько различных форм их представления. В статье выбирается форма, предложенная в [1]. Применение обратного преобразования Фурье совместно с суммированием интегралов по Абелю позволило получить разложения функций плотности многомерных устойчивых распределений (см.[1], [12]). Основным результатом статьи являются представления этих функций с помощью рядов обобщенных функций над пространством Лизоркина. Они позволяют определить порядок убывания главного члена разложения на бесконечности для любого радиального направления. Кроме того, выведенные формулы дают возможность увидеть структуру формирования слагаемых в разложениях. В следствии приводятся примеры для различных случаев носителей спектральной меры многомерных устойчивых законов. The article discusses multidimensional strictly stable distributions. As is known, the density functions of these laws are not represented in closed form, with the exception of the well-known laws of Gauss and Cauchy. Characteristic functions are the starting point for research. There are several different forms of their presentation. The article chooses the form proposed in [1]. The application of the inverse Fourier transform together with the Abel summation of the integrals made it possible to obtain expansions of the density functions of multidimensional stable distributions (see [1], [12]). The main result of the article is the representation of these functions using series of generalized functions over the Lizorkin space. They make it possible to determine the order of decay of the principal term of the expansion at infinity for any radial direction. In addition, the derived formulas make it possible to see the structure of the formation of terms in expansions. In the corollary, examples are given for various cases of the support of the spectral measure of multidimensional stable laws.

Fractional Fourier transform and stability of fractional differential equation on Lizorkin space

In the current study, we conduct an investigation into the Hyers–Ulam stability of linear fractional differential equation using the Riemann–Liouville derivatives based on fractional Fourier transform. In addition, some new results on stability conditions with respect to delay differential equation of fractional order are obtained. We establish the Hyers–Ulam–Rassias stability results as well as examine their existence and uniqueness of solutions pertaining to nonlinear problems. We provide examples that indicate the usefulness of the results presented.

THE CONVOLUTION IN ANISOTROPIC TRIEBEL–LIZORKIN SPACES

In this paper, we investigate the boundedness of the norm of the convolution operator in anisotropic Triebel-Lizorkin spaces. The Triebel-Lizorkin spaces are based on the Lorentz spaces pq L . In the anisotropic case, we take the anisotropic Lorentz space pq L as the base. The main goal of the paper is to solve the following problem: let f and g be functions from some classes of the Triebel-Lizorkin space scale. It is necessary to determine which conditions on the parameters of the spaces from f and g are taken and study which space belongs to their convolution gf  . An analogue of the O'Neil theorem was obtained for the Triebel-Lizorkin space scale αq pτF , where α , τ, p , q are vector parameters. Relations of the form γξ hν βη rμ F F  ↪ αq pτF are obtained, with the corresponding ratios of vector parameters γ βα  , hrp 11 1 1   , νμτ 111  , ηξq 111  . The research method is the functional spaces theory and inequalities of functional and harmonic analysis.

Estimates of damped fractional wave equations

In this paper, for the high frequency part of the solution u(x, t) to the linear fractional damped wave equation, we derive asymptotic-in-time linear estimates in Triebel-Lizorkin spaces. Thus we obtain long time decay estimates in real Hardy spaces Hp for u(x, t). The obtained results are natural extension of the known Lp estimates. Our proof is based on some basic properties of the Triebel-Lizorkin space, as well as an atomic decomposition introduced by Han, Paluszynski and Weiss.

Cauchy representation of fractional Fourier transform for Boehmians

Results relating to fractional Fourier transform and their properties in the Lizorkin space are employed in this paper to investigate the Cauchy representation of fractional Fourier transform for integrable Boehmians. An inversion formula for the fractional Fourier transform is addressed. The conclusion remark of the paper spells the initiation for the present investigation.