The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

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The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

In this paper, we obtain approximation theorems of classes of analytic functions by shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>λ</mi>...

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Bibliographic Details
Main Author:	Antanas Laurinčikas
Format:	Article
Language:	English
Published:	MDPI AG 2025-06-01
Series:	Axioms
Subjects:	Hurwitz zeta-function Lerch zeta-function Mergelyan theorem short intervals universality weak convergence of probability measures
Online Access:	https://www.mdpi.com/2075-1680/14/6/472
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Summary:	In this paper, we obtain approximation theorems of classes of analytic functions by shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>λ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the Lerch zeta-function for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>[</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>+</mo><mi>H</mi><mo>]</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>∈</mo><mo>[</mo><msup><mi>T</mi><mrow><mn>27</mn><mo>/</mo><mn>82</mn></mrow></msup><mo>,</mo><msup><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>]</mo></mrow></semantics></math></inline-formula>. The cases of all parameters, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>,</mo><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, are considered. If the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>log</mi><mrow><mo>(</mo><mi>m</mi><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mo>:</mo><mi>m</mi><mo>∈</mo><msub><mi mathvariant="double-struck">N</mi><mn>0</mn></msub><mo>}</mo></mrow></semantics></math></inline-formula> is linearly independent over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></inline-formula>, then every analytic function in the strip <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>s</mi><mrow><mo>=</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>t</mi></mrow><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mi>σ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo></mrow></semantics></math></inline-formula> is approximated by the above shifts.
ISSN:	2075-1680