Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms
For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2022/6721360 |
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| author | David W. Pravica Njinasoa Randriampiry Michael J. Spurr |
| author_facet | David W. Pravica Njinasoa Randriampiry Michael J. Spurr |
| author_sort | David W. Pravica |
| collection | DOAJ |
| description | For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions Wμ,λt for μ and λ rational with λ>0. These Wμ,λt have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ, the connection of the Wμ,λt to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ, the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W−4/3,1/3t/W−4/3,1/30. A useful set of generalized q-Wallis formulas are developed that play a key role in this study of convergence. |
| format | Article |
| id | doaj-art-ecba916f79a244ffb54832ab136de649 |
| institution | Kabale University |
| issn | 1687-0409 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-ecba916f79a244ffb54832ab136de6492025-02-03T06:12:25ZengWileyAbstract and Applied Analysis1687-04092022-01-01202210.1155/2022/6721360Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier TransformsDavid W. Pravica0Njinasoa Randriampiry1Michael J. Spurr2Department of MathematicsDepartment of MathematicsDepartment of MathematicsFor a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions Wμ,λt for μ and λ rational with λ>0. These Wμ,λt have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ, the connection of the Wμ,λt to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ, the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W−4/3,1/3t/W−4/3,1/30. A useful set of generalized q-Wallis formulas are developed that play a key role in this study of convergence.http://dx.doi.org/10.1155/2022/6721360 |
| spellingShingle | David W. Pravica Njinasoa Randriampiry Michael J. Spurr Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms Abstract and Applied Analysis |
| title | Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms |
| title_full | Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms |
| title_fullStr | Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms |
| title_full_unstemmed | Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms |
| title_short | Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms |
| title_sort | solutions of a class of multiplicatively advanced differential equations ii fourier transforms |
| url | http://dx.doi.org/10.1155/2022/6721360 |
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