Некоторые оценки для обобщенного преобразования Фурье, ассоциированного с оператором Чередника - Опдама

In the classical theory of approximation of functions on $\mathbb{R}^+$, the modulus of smoothness are basically built by means of the translation operators $f \to f(x+y)$. As the notion of translation operators was extended to various contexts (see [2] and [3]), many generalized modulus of smoothness have been discovered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [4] and [5]). In [1], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we also discuss this subject. More specifically, we prove some estimates (similar to those proved in [1]) in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated with the differential-difference operator $T^{(\alpha,\beta)}$ in $L^{2}_{\alpha,\beta}(\mathbb{R})$. For this purpose, we use a generalized translation operator.

On Estimates for the Generalized Fourier-Dunkl Transform in the Space L2

Two useful estimates are proved for the generalized Fourier-Dunkltransform in the space L2 on certain classes of functions characterized by thegeneralized continuity modulus.

Riesz Potential in Generalized Hölder Spaces

The chapter provides an overview of the advanced researches on the multidimensional Riesz potential operator in the generalized Hölder spaces. While being of interest within mathematical modeling in economics, theoretical physics, and other areas of knowledge, the Riesz potential plays a significant role for analysis on fractal sets, and this aspect is briefly outlined. The generalized Hölder spaces provide convenient terminology for formalizing the smoothness concept, which is described here. There are constant and variable order potential type operators considered, including a two-pole spherical one. As a sphere is, in some sense, a convenient set for analysis, there are two results, proved in detail: the conditions for the spherical fractional integral of variable order to be bounded in the generalized Hölder spaces, whose local continuity modulus has a dominant, which may vary from point to point, and the ones for the constant-order two-pole spherical potential type operator.

Functions of Boundedkthp-Variation and Continuity Modulus

A scale of spaces exists connecting the class of functions of boundedkthp-variation in the sense of Riesz-Merentes with the Sobolev space of functions withp-integrablekth derivative. This scale is generated by the generalized functionals of Merentes type. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of smoothness of orderk-1/p.

The least r-concave majorant of the continuity modulus ωr

We define the least r-concave majorant for the modulus of continuity of order r on C[a, b], denoted by −r ω r and we establish the inequality ...

Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces

We consider non-standard Hölder spacesHλ(⋅)(X)of functionsfon a metric measure space (X, d, μ), whose Hölder exponentλ(x) is variable, depending onx∈X. We establish theorems on mapping properties of potential operators of variable orderα(x), from such a variable exponent Hölder space with the exponentλ(x) to another one with a “better” exponentλ(x) +α(x), and similar mapping properties of hypersingular integrals of variable orderα(x) from such a space into the space with the “worse” exponentλ(x) −α(x) in the caseα(x) <λ(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spacesHλ(⋅)(X), but also the generalized Hölder spacesHw(⋅,⋅)(X)of functions whose continuity modulus is dominated by a given functionw(x, h),x∈X, h> 0. We admit variable complex valued ordersα(x), whereℜα(x)may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Hölder spaces with the weightα(x).