Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight
The author constructs orthogonal-type expansions with the simplest possible form for the space \(\mathcal{P}_n\) of orthogonal polynomials of degree \(n\) for a general radially symmetric weight function on \(\mathbb{R}^d\). The main result of this paper is: For any \(d \geq 2\), the explicit construction of a zonal polynomial \(p \in \mathcal{P}_n\) having the following property. Each \(f \in \mathcal{P}_n\) can be reconstructed as a sum of its projections onto the orbit of \(p\) under \(SO(d)\) (symmetries of the weight), and hence of its projections onto the zonal polynomials \(p_{\xi}\) obtained from \(p\) by moving its pole to the set \( \{\xi \in \mathbb{R}^d: |\xi|=1\}\). The approach followed here is first to prove the continuous versions and then to deduce the discrete decompositions. Finally, an application of the continuous tight frame decomposition is presented to obtain a simple closed form for the reproducing kernel of \(\mathcal{P}_n\).
quadrature for trigonometric polynomials
scientific article; zbMATH DE number 5600301
scientific article