Entire Gaussian Functions: Probability of Zeros Absence
In this paper, we consider a random entire function of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,<...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/3/255 |
| _version_ | 1797613471289835520 |
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| author | Andriy Kuryliak Oleh Skaskiv |
| author_facet | Andriy Kuryliak Oleh Skaskiv |
| author_sort | Andriy Kuryliak |
| collection | DOAJ |
| description | In this paper, we consider a random entire function of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>+</mo><mo>∞</mo></mrow></msubsup><msub><mi>ε</mi><mi>n</mi></msub><mrow><mo>(</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>×</mo><msub><mi>ξ</mi><mi>n</mi></msub><mrow><mo>(</mo><msub><mi>ω</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>f</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ε</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is a sequence of independent Steinhaus random variables, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ξ</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula> is such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mover><mo movablelimits="false">lim</mo><mo>¯</mo></mover><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><mroot><mrow><mrow><mo>|</mo></mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow><mi>n</mi></mroot><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>#</mo><mo>{</mo><mi>n</mi><mo lspace="0pt">:</mo><msub><mi>f</mi><mi>n</mi></msub><mo>≠</mo><mn>0</mn><mo>}</mo><mo>=</mo><mo>+</mo><mo>∞</mo><mo>.</mo></mrow></semantics></math></inline-formula> We investigate asymptotic estimates of the probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mo>{</mo><mi>ω</mi><mo lspace="0pt">:</mo><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></mrow></semantics></math></inline-formula> has no zeros inside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi mathvariant="double-struck">D</mi><mo>}</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> outside of some set <i>E</i> of finite logarithmic measure, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><mrow><mi>E</mi><mo>∩</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></msub><mi>d</mi><mi>ln</mi><mi>r</mi><mo><</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions. |
| first_indexed | 2024-03-11T06:56:17Z |
| format | Article |
| id | doaj.art-302ed554fe574271abccdc3af4329506 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T06:56:17Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-302ed554fe574271abccdc3af43295062023-11-17T09:34:56ZengMDPI AGAxioms2075-16802023-03-0112325510.3390/axioms12030255Entire Gaussian Functions: Probability of Zeros AbsenceAndriy Kuryliak0Oleh Skaskiv1Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, UkraineFaculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, UkraineIn this paper, we consider a random entire function of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>+</mo><mo>∞</mo></mrow></msubsup><msub><mi>ε</mi><mi>n</mi></msub><mrow><mo>(</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>×</mo><msub><mi>ξ</mi><mi>n</mi></msub><mrow><mo>(</mo><msub><mi>ω</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>f</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ε</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is a sequence of independent Steinhaus random variables, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ξ</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula> is such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mover><mo movablelimits="false">lim</mo><mo>¯</mo></mover><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><mroot><mrow><mrow><mo>|</mo></mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>|</mo></mrow></mrow><mi>n</mi></mroot><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>#</mo><mo>{</mo><mi>n</mi><mo lspace="0pt">:</mo><msub><mi>f</mi><mi>n</mi></msub><mo>≠</mo><mn>0</mn><mo>}</mo><mo>=</mo><mo>+</mo><mo>∞</mo><mo>.</mo></mrow></semantics></math></inline-formula> We investigate asymptotic estimates of the probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mo>{</mo><mi>ω</mi><mo lspace="0pt">:</mo><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></mrow></mrow></semantics></math></inline-formula> has no zeros inside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi mathvariant="double-struck">D</mi><mo>}</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> outside of some set <i>E</i> of finite logarithmic measure, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><mrow><mi>E</mi><mo>∩</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></msub><mi>d</mi><mi>ln</mi><mi>r</mi><mo><</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.https://www.mdpi.com/2075-1680/12/3/255Gaussian entire functionsSteinhaus entire functionszeros distribution of random entire functions |
| spellingShingle | Andriy Kuryliak Oleh Skaskiv Entire Gaussian Functions: Probability of Zeros Absence Axioms Gaussian entire functions Steinhaus entire functions zeros distribution of random entire functions |
| title | Entire Gaussian Functions: Probability of Zeros Absence |
| title_full | Entire Gaussian Functions: Probability of Zeros Absence |
| title_fullStr | Entire Gaussian Functions: Probability of Zeros Absence |
| title_full_unstemmed | Entire Gaussian Functions: Probability of Zeros Absence |
| title_short | Entire Gaussian Functions: Probability of Zeros Absence |
| title_sort | entire gaussian functions probability of zeros absence |
| topic | Gaussian entire functions Steinhaus entire functions zeros distribution of random entire functions |
| url | https://www.mdpi.com/2075-1680/12/3/255 |
| work_keys_str_mv | AT andriykuryliak entiregaussianfunctionsprobabilityofzerosabsence AT olehskaskiv entiregaussianfunctionsprobabilityofzerosabsence |