Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines.

On the exact evaluation of integrals of wavelets / Pellegrino, E.; Sorgentone, C.; Pitolli, F.. - In: MATHEMATICS. - ISSN 2227-7390. - 11:4(2023), p. 983. [10.3390/math11040983]

On the exact evaluation of integrals of wavelets

Sorgentone C.;Pitolli F.
2023

Abstract

Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines.
2023
wavelet; refinable function; refinement mask; pseudo-spline; wavelet Galerkin method; connection coefficients
01 Pubblicazione su rivista::01a Articolo in rivista
On the exact evaluation of integrals of wavelets / Pellegrino, E.; Sorgentone, C.; Pitolli, F.. - In: MATHEMATICS. - ISSN 2227-7390. - 11:4(2023), p. 983. [10.3390/math11040983]
File allegati a questo prodotto
File Dimensione Formato  
Pellegrino_IntegralsWavelets_2023.pdf

accesso aperto

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Creative commons
Dimensione 397.58 kB
Formato Adobe PDF
397.58 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1674251
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 2
social impact