TY - JOUR
T1 - Fractional and complex pseudo-splines and the construction of Parseval frames
AU - Massopust, Peter
AU - Forster, Brigitte
AU - Christensen, Ole
N1 - Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - 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.
AB - 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.
KW - Filters
KW - Fractional and complex B-splines
KW - Framelets
KW - Parseval frames
KW - Pseudo-splines
KW - Unitary extension principle (UEP)
UR - http://www.scopus.com/inward/record.url?scp=85021903709&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2017.06.023
DO - 10.1016/j.amc.2017.06.023
M3 - Article
AN - SCOPUS:85021903709
SN - 0096-3003
VL - 314
SP - 12
EP - 24
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
ER -