# Analysis of Framelet Transforms on a Simplex

## TL;DR

This paper develops framelet transforms on a 2D simplex, providing exact reconstruction for tight framelets and efficient computation comparable to FFT, advancing analysis tools on simplexes.

## Contribution

It introduces a new construction of framelets on the simplex with fast transform algorithms and theoretical guarantees for tight frames.

## Key findings

- Framelet transforms on the simplex are computationally efficient.
- Exact reconstruction is achieved with tight framelets.
- An explicit example of framelet construction is provided.

## Abstract

In this paper, we construct framelets associated with a sequence of quadrature rules on the simplex $T^{2}$ in $\mathbb{R}^{2}$. We give the framelet transforms -- decomposition and reconstruction of the coefficients for framelets of a function on $T^{2}$. We prove that the reconstruction is exact when the framelets are tight. We give an example of construction of framelets and show that the framelet transforms can be computed as fast as FFT.