On a filter for exponentially localized kernels based on Jacobi polynomials

Let α,β≥-12, and for k=0,1,…k=0,1,…, pk(α,β) denote the orthonormalized Jacobi polynomial of degree kk. We discuss the construction of a matrix HH so that there exist positive constants cc, c1c1, depending only on HH, αα, and ββ such that ∑k=0∞Hk,npk(α,β)(cosθ)pk(α,β)(cosϕ)≤c1n2max(α,β)+2exp(-cn(θ-ϕ)2),θ,ϕ∈[0,π],n=1,2,….Specializing to the case of Chebyshev polynomials, α=β=-12, we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L2L2 space.