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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2025-06-01
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| author | Antanas Laurinčikas |
| author_facet | Antanas Laurinčikas |
| author_sort | Antanas Laurinčikas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e0475ef026d24259838f2d36fd1e58e1 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-e0475ef026d24259838f2d36fd1e58e12025-08-20T02:24:35ZengMDPI AGAxioms2075-16802025-06-0114647210.3390/axioms14060472The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short IntervalsAntanas Laurinčikas0Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, LithuaniaIn 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.https://www.mdpi.com/2075-1680/14/6/472Hurwitz zeta-functionLerch zeta-functionMergelyan theoremshort intervalsuniversalityweak convergence of probability measures |
| spellingShingle | Antanas Laurinčikas The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals Axioms Hurwitz zeta-function Lerch zeta-function Mergelyan theorem short intervals universality weak convergence of probability measures |
| title | The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals |
| title_full | The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals |
| title_fullStr | The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals |
| title_full_unstemmed | The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals |
| title_short | The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals |
| title_sort | approximation of analytic functions using shifts of the lerch zeta function in short intervals |
| topic | Hurwitz zeta-function Lerch zeta-function Mergelyan theorem short intervals universality weak convergence of probability measures |
| url | https://www.mdpi.com/2075-1680/14/6/472 |
| work_keys_str_mv | AT antanaslaurincikas theapproximationofanalyticfunctionsusingshiftsofthelerchzetafunctioninshortintervals AT antanaslaurincikas approximationofanalyticfunctionsusingshiftsofthelerchzetafunctioninshortintervals |