Exponential Splines of Complex Order

We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order. We derive a time domain representation of a complex exponential B-spline depending on a single parameter and establish a connection to fractional differential operators defined on Lizorkin spaces. Moreover, we prove that complex exponential splines give rise to multiresolution analyses of $L^2(\mathbb{R})$ and define wavelet bases for $L^2(\mathbb{R})$

Fractional Operators, Dirichlet Averages, and Splines

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented

Some remarks about the connection between fractional divided differences, fractional B-splines, and the Hermite-Genocchi formula.

Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite-Genochi formula. This formula then allows the definition of a larger class of fractional B-splines

SOME REMARKS ABOUT THE CONNECTION BETWEEN FRACTIONAL DIVIDED DIFFERENCES, FRACTIONAL B-SPLINES, AND THE HERMITE-GENOCCHI FORMULA (SHORT PAPER)

Communicated by (xxxxxxxxxx) Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite-Genochi formula. This formula then allows the definition of a larger class of fractional B-splines