Abstract
In this paper the perturbed system of exponents with some asymptotics is considered. Basis properties of this system in Lebesgue spaces with variable summability exponent are investigated.
Highlights
Consider the following system of exponents: eiλnt n∈Z, ( )where {λn} ⊂ R is a sequence of real numbers, Z is a set of integer numbers
Where {λn} ⊂ R is a sequence of real numbers, Z is a set of integer numbers
It is the aim of this paper to investigate basis properties of the system ( ) in Lebesgue space Lpt with variable summability index p(t), when {λn} has the asymptotics λn = n – α sign n + O |n|–β, n → ∞, ( )
Summary
Where {λn} ⊂ R is a sequence of real numbers, Z is a set of integer numbers It is the aim of this paper to investigate basis properties (basicity, completeness, and minimality) of the system ( ) in Lebesgue space Lpt with variable summability index p(t), when {λn} has the asymptotics λn = n – α sign n + O |n|–β , n → ∞, where α, β ∈ R are some parameters. Lemma Let X be a Banach space with basis {xn}n∈N ⊂ X and F : X → X be a Fredholm operator. Theorem (Krein-Milman-Rutman [ ]) Let X be a Banach space with norm · , {xn}n∈N ⊂ X be normed basis in X (i.e. xn = , ∀n ∈ N ) with conjugate system {x∗n}n∈N ⊂ X∗, and {yn}n∈N ⊂ X be a system satisfying the inequality xn – yn < γ – , n=.
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