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On the Exact Evaluation of Integrals of Wavelets

Author

Listed:
  • Enza Pellegrino

    (Department of Industrial and Information Engineering and Economics, University of L’Aquila, E. Pontieri 2, 67040 Roio Poggio, Italy
    These authors contributed equally to this work.)

  • Chiara Sorgentone

    (Department of Basic and Applied Sciences for Engineering, Università di Roma “La Sapienza”, Via Antonio Scarpa 16, 00161 Roma, Italy
    These authors contributed equally to this work.)

  • Francesca Pitolli

    (Department of Basic and Applied Sciences for Engineering, Università di Roma “La Sapienza”, Via Antonio Scarpa 16, 00161 Roma, Italy
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Enza Pellegrino & Chiara Sorgentone & Francesca Pitolli, 2023. "On the Exact Evaluation of Integrals of Wavelets," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:983-:d:1068912
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