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A universal formula for generalized cardinal B-splines

Amini, A ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.acha.2016.10.004
  3. Publisher: Academic Press Inc
  4. Abstract:
  5. We introduce a universal and systematic way of defining a generalized B-spline based on a linear shift-invariant (LSI) operator L (a.k.a. Fourier multiplier). The generic form of the B-spline is . βL=LdL-1δ where . L-1δ is the Green's function of L and where . Ld is the discretized version of the operator that has the smallest-possible null space. The cornerstone of our approach is a main construction of . Ld in the form of an infinite product that is motivated by Weierstrass' factorization of entire functions. We show that the resulting Fourier-domain expression is compatible with the construction of all known B-splines. In the special case where L is the derivative operator (linked with piecewise-constant splines), our formula is equivalent to Euler's celebrated decomposition of . sinc(x)=sin (πx)πx into an infinite product of polynomials. Our main challenge is to prove convergence and to establish continuity results for the proposed infinite-product representation. The ultimate outcome is the demonstration that the generalized B-spline . βL generates a Riesz basis of the space of cardinal L-splines, where L is an essentially arbitrary pseudo-differential operator
  6. Keywords:
  7. Differential operator ; Spline ; Transfer function ; Distributed parameter control systems ; Interpolation ; Mathematical operators ; Derivative operators ; Discrete approximation ; Fourier multipliers ; Linear shift invariants ; Piece-wise constants ; Pseudo-differential operator ; Universal formulas ; Ship propellers
  8. Source: Applied and Computational Harmonic Analysis ; 2016 ; 10635203 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S1063520316300707