$h$-Trigonometric B-splines

TL;DR

This paper introduces discrete analogues of exponential, sine, and cosine functions, and develops trigonometric B-splines with recurrence relations, derivatives, and identities, extending classical B-spline theory to a discrete trigonometric setting.

Contribution

It defines discrete trigonometric B-splines and derives fundamental properties, extending classical polynomial B-spline results to the discrete trigonometric context.

Findings

Derived a two-term recurrence relation for discrete trigonometric B-splines.

Established a two-term formula for the discrete derivative.

Presented variants of the Marsden identity for these splines.

Abstract

We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a two-term recurrence relation, a two-term formula for the discrete derivative, and two variants of the Marsden identity for these discrete trigonometric B-splines. Since the classical exponential, sine, and cosine functions are limiting cases of their discrete analogues, we conclude that many of the standard results for classical polynomial B-splines extend naturally both to trigonometric B-splines and to discrete trigonometric B-splines.