IDEAS home Printed from https://ideas.repec.org/a/wly/jnlaaa/v2022y2022i1n6721360.html

Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

Author

Listed:
  • David W. Pravica
  • Njinasoa Randriampiry
  • Michael J. Spurr

Abstract

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−4313/,/t/W−4313/,/0. A useful set of generalized q‐Wallis formulas are developed that play a key role in this study of convergence.

Suggested Citation

  • David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2022. "Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jnlaaa:v:2022:y:2022:i:1:n:6721360
    DOI: 10.1155/2022/6721360
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2022/6721360
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2022/6721360?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. A. Lastra & S. Malek, 2014. "On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. A. Lastra & S. Malek, 2014. "On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-29, December.
    3. Steven G. Krantz, 1999. "Handbook of Complex Variables," Springer Books, Springer, number 978-1-4612-1588-2, October.
    4. Alberto Lastra & Stéphane Malek, 2013. "On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-18, November.
    5. Alberto Lastra & Stéphane Malek, 2013. "On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    6. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2014. "Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-24, October.
    7. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2014. "Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2024. "Solutions of Inhomogeneous Multiplicatively Advanced ODEs and PDEs with a q‐Fredholm Theory and Applications to a q‐Advanced Schrödinger Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2024(1).
    2. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2014. "Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnlaaa:v:2022:y:2022:i:1:n:6721360. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4058 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.