On q-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet

Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J.

doi:10.1016/j.acha.2016.05.003

Cited by 11 publications

(25 citation statements)

References 24 publications

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“…In various theories of differential equations, convolutions provide a useful tool since general solutions can be determined from fundamental solutions, as demonstrated here in Equation (33). This is one motivation for obtaining solutions to homogenous equations, as appears in Proposition 2.…”

Section: Brief Overviewmentioning

confidence: 99%

“…The introduction of new functions are as follows: For K(t) see [32] for decay constants K 1 and K 2 in Table 1; The functions q Cos(t) and q Sin(t) are closely related to Cq (t) and Sq (t), respectively, introducted in [25], where constants K 3 and K 4 are obtained; The q-Bessel functions, related to J (t), were introduced in [33], along with decay constants K 5 and K 6 ; Flat wavelets F(t) have Fourier transforms that are averages of theta functions, first derived in [29], along with constants K 7 and K 8 . The functions K * Cq (t) and W µ,λ (t), have Fourier transforms that involve theta functions, which can be used to obtain decay parameters K 9 and K 10 .…”

Section: Expanded Table Of Fourier Transformsmentioning

confidence: 99%

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Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

Pravica

Randriampiry

Spurr

2020

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A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( δ ) ( t ) = E W ( q γ t ) where the eigenvalue E ∈ R is independent of the advancing parameter q > 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.

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Section: Brief Overviewmentioning

confidence: 99%

Section: Expanded Table Of Fourier Transformsmentioning

confidence: 99%

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

Pravica

Randriampiry

Spurr

2020

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“…For certain low values of δ, we show that the W μ,λ ðtÞ give unique, canonical extensions of the associated f μ,λ ðtÞ which satisfy the MADE (2). In general, for higher values of δ, the W μ,λ ðtÞ are not extensions of f μ,λ ðtÞ; however, they are all Schwartz wavelets with connections not only to wavelet theory but also to special function theory in that F½W μ,λ ðtÞðxÞ is expressed in terms of (6) and thus in terms of the Jacobi theta function. See (98) and (99) of Theorem 13 for specifics.…”

Section: Introductionmentioning

confidence: 99%

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

Pravica

Randriampiry

Spurr

2022

Abstract and Applied Analysis

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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 W μ , λ t for μ and λ rational with λ > 0 . These W μ , λ t have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ , the connection of the W μ , λ t 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 / 3 t / W − 4 / 3 , 1 / 3 0 . A useful set of generalized q -Wallis formulas are developed that play a key role in this study of convergence.

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“…The study of q-difference equations has also been under study in different applications in the last years. Some advances in this respect are [14][15][16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients

2019

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We consider analytic and formal solutions of certain family of q-difference-differential equations under the action of a complex perturbation parameter. The previous study (Lastra and Malek in Adv. Differ. Equ. 2015:344, 2015) provides information in the case where the main equation under study is factorizable as a product of two equations in the so-called normal form. Each of them gives rise to a single level of q-Gevrey asymptotic expansion. In the present work, the main problem under study does not suffer any factorization, and a different approach is followed. More precisely, we lean on the technique developed in (Dreyfus in Int. Math. Res. Not. 15:6562-6587, 2015, where the first author makes distinction among the different q-Gevrey asymptotic levels by successive applications of two q-Borel-Laplace transforms of different orders, both to the same initial problem, which can be described by means of a Newton polygon.

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On q-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet

Cited by 11 publications

References 24 publications

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

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

On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients

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