# H\"olderian convergence of fractional extended nabla operator to fractional derivative

## TL;DR

This paper introduces a fractional extended nabla operator based on linear spline backward differences, proves its strong convergence to fractional derivatives in H"older spaces, and supports findings with numerical examples.

## Contribution

It constructs a new fractional operator as a power of a spline-based difference operator and proves its convergence to fractional derivatives.

## Key findings

- Operator converges strongly to fractional derivatives in H"older spaces
- Numerical examples validate theoretical convergence
- Provides a new approach to fractional difference operators

## Abstract

In this paper, we construct the fractional extended nabla operator as fractional power of linear spline of backward difference operator. Then we prove the strong convergence of this operator to fractional derivative in a H\"older space setting. Finally numerical examples are presented.