On the Periodized Square of L² Cardinal Splines

Abstract

We establish properties of and propose a conjecture concerning ∑ _m (S(x+m))², where S is a piecewise polynomial cardinal spline in .

2000 AMS Subject Classification:

• 65A15
• 62G07

Keywords:

• Euler exponential splines
• extremes
• piecewise polynomial splines
• symmetric polynomial expansions

ACKNOWLEDGMENTS

To establish Corollary 1.2 in the case of odd n, we used a property concerning the maximum of the modulus of the so-called Euler exponential splines. These splines were christened and studied by Schoenberg; see, for example, [Schoenberg 73, Schoenberg 83]. The authors would like to thank Carl de Boor, who kindly responded to a query concerning this matter and provided the reference [de Boor 76], which contains a proof of this property.