Legendre polynomials Triple Product Integral and lower-degree approximation of polynomials using Chebyshev polynomials

Abstract

In this report, we present two mathematical results which can be useful in a variety of settings. First, we present an analysis of Legendre polynomials triple product integral. Such integrals arise whenever two functions are multiplied, with both the operands and the result represented in the Legendre polynomial basis. We derive a recurrence relation to calculate these integrals analytically. We also establish the sparsity of the triple product integral tensor, and derive the Legendre polynomial triple product integral theorem, giving the exact closed form expression for the sparsity structure. Secondly, we derive a truncation scheme to approximate a polynomial with a lower degree polynomial, while keeping the approximation error low under the L_{infty} norm. We use the Chebyshev polynomials to derive our truncation scheme. We present empirical results which suggest that the approximation error is quite low, even for fairly low degree approximations.