On the zeros of the generalized Euler-Frobenius Laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples

[摘要] Let h(t) be a nonnegative measurable function supported in [-1/2, 1/2] and M-d(t) = (chi([-1/2, 1/2])star chi([-1/2, 1/2]) star ... star chi([-1/2, 1/2]))(t) (d + 1 times) be the central B-spline of degree d. We show that the roots of the generalized Euler-Frobenius Laurent polynomial defined by Sigma(h,d)(z) := Sigma(n is an element of Z)(h star M-d)(n)z(n) are simple, negative and all are different from -1. As. a consequence of this result, we show that for every sequence {y(n)}(n is an element of Z) is an element of R-Z of samples having polynomial growth and nonnegative measurable function h supported in [-1/2, 1/2], there is a unique spline f of degree d with polynomial growth satisfying (f star h)(n) = y(n), n is an element of Z. The presented work answers affirmatively the open problem posed in Perez-Villalon and Portal (2012) [9]. (C) 2015 Published by Elsevier Inc.

[发布日期] 2015-12-15

[关键词] Spline interpolation;Generalized Euler-Frobenius;Laurent polynomial;Average sampling