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.

Characterization of Dini Lipschitz Functions in Terms of Their Helgason Transform

In this paper, using a generalized translation operator, we obtain an analog of Younis Theorem 5.2 in [6] for the Helgason Fourier transform of a set of functions satisfying the Dini Lipschitz condition in the space L2 for functions on noncompact rank one Riemannian symmetric spaces.

Characterization of Dini–Lipschitz functions for the Helgason Fourier transform on rank one symmetric spaces

In this paper, using a generalized translation operator, we obtain an analog of Younis’ theorem, [

Bessel Transform of -Bessel Lipschitz Functions

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the -Bessel Lipschitz condition in .