Complex B-splines and Hurwitz zeta functions

Abstract

We characterize nonempty open subsets of the complex plane where the sum ζ(s, α) + e ±iπs ζ(s, 1 − α) of Hurwitz zeta functions has no zeros in s for all 0 α 1. This problem is motivated by the construction of fundamental cardinal splines of complex order s.

Fractional Operators, Dirichlet Averages, and Splines

“…Using Dirichlet averages, several classical multivariate B-spline identities were generalized to the complex setting. The existence of fundamental complex B-splines was investigated in [17] and in [16] periodic complex B-splines were constructed. There also exist interesting relationships between complex B-splines, Dirichlet averages and difference operators, several of which are highlighted in [12].…”

“…For z = n ∈ N 0 , Eqns. (19), (20), and (up to a factor (−1) n ) (21) reduce to the standard forms (16), (17), and (18) for nth order finite differences on uniform knots and classical B-splines. We recall the following relation between the n-th order cardinal B-spline B n , n ∈ N, and the divided differences:…”

Interpolation and Sampling with Exponential Splines of Real Order

“…As it turns out, even generalizations of these polynomial B-splines, namely, polynomial B-splines of complex and even quaternionic order, do possess associated fundamental splines provided the order is chosen to lie in certain nonempty subregions of the complex plane or quaternionic space. For details, we refer the interested reader to [7] in the former case and to [11] in the latter.…”

Interpolation and Sampling with Exponential Splines of Real Order

“…As it turns out, even generalizations of these polynomial B-splines, namely, polynomial B-splines of complex and even quaternionic order, do possess associated fundamental splines provided the order is chosen to lie in certain nonempty subregions of the complex plane or quaternionic space. For details, we refer the interested reader to [7] in the former case and to [10] in the latter.…”