A Universal Formula for Generalized Cardinal BSplines
A. Amini, R. Madani, M. Unser
Applied and Computational Harmonic Analysis, vol. 45, no. 2, pp. 341358, September 2018.
We introduce a universal and systematic way of defining a generalized Bspline based on a linear shiftinvariant (LSI) operator L (a.k.a. Fourier multiplier). The generic form of the Bspline is β_{L} = L_{d} L^{−1} δ where L^{−1} δ is the Green's function of L and where L_{d} is the discretized version of the operator that has the smallestpossible null space. The cornerstone of our approach is a main construction of L_{d} in the form of an infinite product that is motivated by Weierstrass' factorization of entire functions. We show that the resulting Fourierdomain expression is compatible with the construction of all known Bsplines. In the special case where L is the derivative operator (linked with piecewiseconstant splines), our formula is equivalent to Euler's celebrated decomposition of sinc(x) = sin(π x) ∕ (π x) into an infinite product of polynomials. Our main challenge is to prove convergence and to establish continuity results for the proposed infiniteproduct representation. The ultimate outcome is the demonstration that the generalized Bspline β_{L} generates a Riesz basis of the space of cardinal Lsplines, where L is an essentially arbitrary pseudodifferential operator.

@ARTICLE(http://bigwww.epfl.ch/publications/amini1801.html,
AUTHOR="Amini, A. and Madani, R. and Unser, M.",
TITLE="A Universal Formula for Generalized Cardinal \mbox{{B}Splines}",
JOURNAL="Applied and Computational Harmonic Analysis",
YEAR="2018",
volume="45",
number="2",
pages="341358",
month="September",
note="")
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