Approximation by interpolation spectral subspaces of operators with discrete spectrum

Abstract

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces

“…The inequalities (12) and ( 13) follow from the inequalities (10) and (11) correspondingly, taking into account that N ϑ,2 = ((2/π) sin(πϑ)) 1/2 (see [14,Exercise B.5]) and ϑ = 1/(s + 1).…”

“…Investigation of the relation between the smoothness properties of functions and the possible orders of their approximations were carried out by many authors on various classes of functions [5,6]. These results are extended to approximations in a Banach space, in particular, to spectral approximations for a linear closed unbounded operator [7,10,12]. In this case, special approximation scales of analytic vectors of finite exponential types and abstract Besov-type approximation spaces are considered.…”