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

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.
Scheda breve Scheda completa
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]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11573/1674251`