Basicity of a Perturbed System of Exponents in Lebesgue Spaces With a Variable Summability Exponent

Abstract

In this paper a system of exponents 1 is considered, where is some polynomial of degree m, m ∈ N. It is proved that under certain conditions on the exponent p (·) the basicity of this system in a Lebesgue space with a variable summability exponent Lp(·) (−π, π) depends on the coefficient αm−1 and m. Moreover, in the case of basicity, it is isomorphic to the classical system of exponents in Lp(·) (−π, π). Earlier in the case the Riesz basicity of this system in L2 (−π, π) was established by Yu.A.Kazmin

Basicity of a perturbed system of exponents in rearrangement invariant spaces

“…It should be noted that a result similar to this corollary for the Lebesgue space L p(•) (−π, π) with variable summability exponent was obtained in [27].…”

“…A similar result for the Lebesgue space with a variable summability exponent was obtained in [12], the weighted case of the space was considered in [25,26]. In [27], a sufficient condition for the basicity of the system E λ in Lebesgue spaces with a variable summability exponent is found. Similar problems have been also studied in [2,17,18,30].…”

“…We will consider the case when the coefficients a k , k = 0, m − 1, are real. As shown in [27], the following asymptotic formula holds…”