Further Properties of Quantum Spline Spaces

Gülter Budakçı,Halil Oruç

doi:10.3390/math8101770

Abstract

We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.

Highlights

B-splines were constructed by Lobachevsky as convolutions of certain probability distributions in the early 19th century
Mangasarian and Schumaker introduced discrete splines, h-splines, to solve discrete analogoues of minimization problems in a Banach space. These discrete splines of degree n are defined on a subset of real line of the form [ a, b]h = { a, a + h, . . . , a + Nh}, b = a + Nh, whose knot sequence is in [ a, b] and their polynomial pieces agree at the knots up to the order n − 1 of forward differences with step size h instead of derivatives
In [6], fundamental formulas of classical B-splines were extended to q-B-splines

Summary

Introduction

B-splines were constructed by Lobachevsky as convolutions of certain probability distributions in the early 19th century. Mangasarian and Schumaker introduced discrete splines, h-splines, to solve discrete analogoues of minimization problems in a Banach space (see [4]) These discrete splines of degree n are defined on a subset of real line of the form [ a, b]h = { a, a + h, . The q-splines are piecewise polynomials whose q-derivatives up to some order agree at the joins. A recent study relates q-B-splines with the q-Peano kernels of divided differences and solves a best approximation problem in the space of quantum integrable functions, see [7]. S is a continuous piecewise polynomial of degree at most n and quantum continuous of order n − 1 at the knots. Syntax Warning: Mismatch between font type and embedded font file urthermore, we find the polynomial pieces of a q-spline by using quantum derivatives

Preliminaries

Properties of q-B-Splines

Conclusions