On basicity of perturbed exponential system with piecewise linear phase in Morrey-type spaces
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On basicity of perturbed exponential system with piecewise linear phase in Morrey-type spaces Bilal Bilalov1
· Togrul Muradov1 · Fidan Seyidova2
Received: 29 December 2018 / Accepted: 18 July 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract An exponential system with piecewise linear phase depending on some parameters is considered in this work. Basis properties of this system (such as completeness, minimality and basicity) are studied in a subspace of Morrey space where continuous functions are dense. A sufficient condition for the completeness (minimality or basicity) of this system in the mentioned subspace is found. Keywords Exponential system · Basicity · Morrey space Mathematics Subject Classification 33B10 · 46E30 · 54D70
1 Introduction Basis properties of the exponential system E β ≡ ei(nt+ β|t| sign n)
n∈Z
(1)
are studied in this work, where β ∈ C is a complex parameter, and Z is the set of integers. This system is a modification of the following perturbed exponential system: , eβ ≡ ei(n+βsign n)t n∈Z
which has been considered by many mathematicians. The study of basis properties of eβ (such as completeness, minimality and basicity) has a long history. It dates back to the works by Paley–Wiener [1] and Levinson [2,3]. Basicity (Riesz basicity) criterion for the system eβ in L 2 (−π, π) with respect to the real parameter β ∈ R follows from the results obtained by Levinson [2,3] and Kadets [4], and this criterion is the inequality |β| < 41 . Basicity criterion for the system eβ in the Lebesgue spaces L p (−π, π) , 1 < p < +∞, with respect to the
B
Bilal Bilalov [email protected]
1
Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9, B.Vahabzade Str., AZ 1141 Baku, Azerbaijan
2
Ganja State University, Ganja, Azerbaijan
123
B. Bilalov et al.
parameter β has been obtained later by Sedletski [5] and Moiseev [6]. Basis properties of eβ are closely related to the similar properties of perturbed sine systems {sin (n − β) t}n∈N ,
(2)
and cosine systems 1 ∪ {cos (n − β) t}n∈N ,
(3)
{cos (n − β) t}n∈Z + , (Z + = {0} ∪ N )
(4)
in corresponding Banach spaces of functions on [0, π]. These systems arise when solving partial differential equations of mixed (or elliptic) type by Fourier method in special domains. To justify the formally constructed solution, it is very important to study the basis properties of these systems in appropriate spaces of functions (see, e.g., [7,8]). Many authors have studied the basis properties of systems in various functional spaces (mainly Lebesgue spaces and their weighted versions; see, e.g., [9–22,38]). The works which consider the approximation properties of the systems (1)–(4) can be divided into two groups. The first one includes the works which used the methods of the theory of entire functions (see, e.g., [1–6]), and the second group consists of those which used the methods of boundary value problems for analytic functions (see, e.g., [7,9,21,39]). The latter idea originated f
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