Fractional and complex pseudo-splines and the construction of Parseval frames

Peter Massopust, Brigitte Forster, Ole Christensen

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

Pseudo-splines of integer order (m, ℓ) were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies’ scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders (z, ℓ) with α ≔ Re z ≥ 1. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from Cm, m∈N0, one uses a continuous family of functions belonging to the Hölder spaces Cα−1. The presence of the imaginary part of z allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle. The regularity and approximation order of this new class of generalized splines is also discussed.
Original languageEnglish
JournalApplied Mathematics and Computation
Volume314
Pages (from-to)12-24
ISSN0096-3003
DOIs
Publication statusPublished - 2017

Keywords

  • Numerical Methods
  • Filters
  • Fractional and complex B-splines
  • Framelets
  • Parseval frames
  • Pseudo-splines
  • Unitary extension principle (UEP)
  • Computational methods
  • Filters (for fluids)
  • Mathematical techniques
  • Approximation orders
  • B splines
  • Extension principles
  • Imaginary parts
  • Increased flexibility
  • Scaling functions
  • Interpolation

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