Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>τ</mi></msub></semantics></math></inline-formula> be a solution to the equation <inline-...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/11/613 |
| _version_ | 1797469043818496000 |
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| author | Antanas Laurinčikas |
| author_facet | Antanas Laurinčikas |
| author_sort | Antanas Laurinčikas |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>τ</mi></msub></semantics></math></inline-formula> be a solution to the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>τ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>π</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the increment of the argument of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>π</mi><mrow><mo>−</mo><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup><mi mathvariant="sans-serif">Γ</mi><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> along the segment connecting points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>τ</mi></msub></semantics></math></inline-formula> is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>τ</mi><msub><mi>α</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>τ</mi><msub><mi>α</mi><mi>r</mi></msub></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>α</mi><mi>r</mi></msub></mrow></semantics></math></inline-formula> are different positive numbers, in the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>+</mo><mi>H</mi><mo>]</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Finally, an example of the approximation of analytic functions by a composition of the above shifts is given. |
| first_indexed | 2024-03-09T19:15:52Z |
| format | Article |
| id | doaj.art-1a72dfc33aa546758045ec704ebcb276 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T19:15:52Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-1a72dfc33aa546758045ec704ebcb2762023-11-24T03:44:17ZengMDPI AGAxioms2075-16802022-11-01111161310.3390/axioms11110613Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function IIAntanas Laurinčikas0Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, LithuaniaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>τ</mi></msub></semantics></math></inline-formula> be a solution to the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>τ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>π</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the increment of the argument of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>π</mi><mrow><mo>−</mo><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup><mi mathvariant="sans-serif">Γ</mi><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> along the segment connecting points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>τ</mi></msub></semantics></math></inline-formula> is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>τ</mi><msub><mi>α</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>τ</mi><msub><mi>α</mi><mi>r</mi></msub></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>α</mi><mi>r</mi></msub></mrow></semantics></math></inline-formula> are different positive numbers, in the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>+</mo><mi>H</mi><mo>]</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Finally, an example of the approximation of analytic functions by a composition of the above shifts is given.https://www.mdpi.com/2075-1680/11/11/613Gram functionjoint universalityRiemann zeta-functionweak convergence |
| spellingShingle | Antanas Laurinčikas Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II Axioms Gram function joint universality Riemann zeta-function weak convergence |
| title | Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II |
| title_full | Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II |
| title_fullStr | Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II |
| title_full_unstemmed | Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II |
| title_short | Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II |
| title_sort | joint approximation of analytic functions by shifts of the riemann zeta function twisted by the gram function ii |
| topic | Gram function joint universality Riemann zeta-function weak convergence |
| url | https://www.mdpi.com/2075-1680/11/11/613 |
| work_keys_str_mv | AT antanaslaurincikas jointapproximationofanalyticfunctionsbyshiftsoftheriemannzetafunctiontwistedbythegramfunctionii |