Abstract
The perturbed system of exponents with a piecewise linear phase, consisting of eigenfunctions of a discontinuous differential operator, is considered in this work. Under certain conditions on the weight function of the form of a power function, sufficient conditions for the basicity of this system are obtained in generalized weighted Lebesgue space.
Highlights
1 Introduction The perturbed system of exponents {eiλnt}n∈Z plays an important role in the study of spectral properties of discrete differential operators and in the approximation theory
There arose a great interest in considering various problems, related to some research fields of mechanics and mathematical physics, in generalized Lebesgue spaces Lp(·) with a variable summability exponent p(·)
It should be noted that the application of the Fourier method to the problems for partial differential equations in generalized Sobolev classes requires a good knowledge of the approximative properties of perturbed exponential systems in generalized Lebesgue spaces
Summary
The perturbed system of exponents {eiλnt}n∈Z plays an important role in the study of spectral properties of discrete differential operators and in the approximation theory. There arose a great interest in considering various problems, related to some research fields of mechanics and mathematical physics, in generalized Lebesgue spaces Lp(·) with a variable summability exponent p(·). It should be noted that the application of the Fourier method to the problems for partial differential equations in generalized Sobolev classes requires a good knowledge of the approximative properties of perturbed exponential systems in generalized Lebesgue spaces. When the weight of the form of power function, the basis properties of this system are studied in a weighted space Lp(·),ρ . Let us present some facts from the theory of Lebesgue spaces with a variable summability exponent. Property B If p(t) : < p– ≤ p+ < +∞, the class C ∞(–π , π ) (class of finite and indefinitely differentiable functions) is everywhere dense in Lp(·)
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