Bessel type transform associated with Titchmarsh’s Theorem

In this paper we have extended Titchmarsh’s theorem for the Bessel type transform for function on half-line [0, ∞) in a weighted Lp− metric are studied with the use of Bessel type generalized translation

G-frame representations with bounded operators

Dynamical sampling, as introduced by Aldroubi et al., deals with frame properties of sequences of the form [Formula: see text], where [Formula: see text] belongs to Hilbert space [Formula: see text] and [Formula: see text] belongs to certain classes of bounded operators. Christensen et al. studied frames for [Formula: see text] with index set [Formula: see text] (or [Formula: see text]), that has representations in the form [Formula: see text] (or [Formula: see text]). As frames of subspaces, fusion frames and generalized translation invariant systems are the special cases of [Formula: see text]-frames, the purpose of this paper is to study and get sufficient conditions for [Formula: see text]-frames [Formula: see text] (or [Formula: see text] having the form [Formula: see text] [Formula: see text] (or [Formula: see text] [Formula: see text]). Also, we get the relation between representations of dual [Formula: see text]-frames with index set [Formula: see text]. Finally, we study stability of [Formula: see text]-frame representations under some perturbations.

General Euler-Poisson-Darboux Equation and Hyperbolic B-Potentials

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

On the K-functional in learning theory

[Formula: see text]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [Formula: see text]-functionals in [Formula: see text] spaces with [Formula: see text]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [Formula: see text]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.

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.