Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mi>j</mi></msub...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3465 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1846153095527006208 |
|---|---|
| author | Ayed. R. A. Alanzi Shokrya S. Alshqaq Raouf Fakhfakh |
| author_facet | Ayed. R. A. Alanzi Shokrya S. Alshqaq Raouf Fakhfakh |
| author_sort | Ayed. R. A. Alanzi |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mi>j</mi></msub></mrow><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mi>j</mi></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>)</mo></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula>, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>2</mn></msub></semantics></math></inline-formula>. Introduce the set of measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>⊞</mo><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>⊞</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mn>1</mn></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mrow><mi>a</mi><mi>n</mi><mi>d</mi></mrow><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mn>2</mn></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>)</mo></mrow><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> We show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> remains a CSK family, (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi mathvariant="fraktur">F</mi><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is a non-degenerate compactly supported measure), then the measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>2</mn></msub></semantics></math></inline-formula> are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied. |
| format | Article |
| id | doaj-art-b6fec098339845668a9493bd6595ce7a |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-b6fec098339845668a9493bd6595ce7a2024-11-26T18:11:31ZengMDPI AGMathematics2227-73902024-11-011222346510.3390/math12223465Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions ProductAyed. R. A. Alanzi0Shokrya S. Alshqaq1Raouf Fakhfakh2Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi ArabiaDepartment of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi ArabiaDepartment of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi ArabiaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mi>j</mi></msub></mrow><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mi>j</mi></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mi>j</mi></msub></msubsup><mo>)</mo></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula>, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>2</mn></msub></semantics></math></inline-formula>. Introduce the set of measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>⊞</mo><mi mathvariant="fraktur">F</mi><mrow><mo>(</mo><msub><mi>ν</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>⊞</mo><msubsup><mi>Q</mi><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>,</mo><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mn>1</mn></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mspace width="4pt"></mspace><mspace width="4pt"></mspace><mrow><mi>a</mi><mi>n</mi><mi>d</mi></mrow><mspace width="4pt"></mspace><mspace width="4pt"></mspace><msub><mi>m</mi><mn>2</mn></msub><mo>∈</mo><mrow><mo>(</mo><msubsup><mi>m</mi><mo>−</mo><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>,</mo><msubsup><mi>m</mi><mo>+</mo><msub><mi>ν</mi><mn>2</mn></msub></msubsup><mo>)</mo></mrow><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> We show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> remains a CSK family, (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi mathvariant="fraktur">F</mi><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is a non-degenerate compactly supported measure), then the measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mn>2</mn></msub></semantics></math></inline-formula> are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.https://www.mdpi.com/2227-7390/12/22/3465variance functionfree and Boolean convolutionsCauchy–Stieltjes transformMarchenko–Pastur law |
| spellingShingle | Ayed. R. A. Alanzi Shokrya S. Alshqaq Raouf Fakhfakh Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product Mathematics variance function free and Boolean convolutions Cauchy–Stieltjes transform Marchenko–Pastur law |
| title | Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product |
| title_full | Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product |
| title_fullStr | Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product |
| title_full_unstemmed | Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product |
| title_short | Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product |
| title_sort | stability of cauchy stieltjes kernel families by free and boolean convolutions product |
| topic | variance function free and Boolean convolutions Cauchy–Stieltjes transform Marchenko–Pastur law |
| url | https://www.mdpi.com/2227-7390/12/22/3465 |
| work_keys_str_mv | AT ayedraalanzi stabilityofcauchystieltjeskernelfamiliesbyfreeandbooleanconvolutionsproduct AT shokryasalshqaq stabilityofcauchystieltjeskernelfamiliesbyfreeandbooleanconvolutionsproduct AT raouffakhfakh stabilityofcauchystieltjeskernelfamiliesbyfreeandbooleanconvolutionsproduct |