On a filter for exponentially localized kernels based on Jacobi polynomials

Let α, β ≥ - frac(1, 2), and for k = 0, 1, ..., p_k^{(α, β)} denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c₁, depending only on H, α, and β such that fenced(underover(∑, k = 0, ∞) H_{k, n} p_k^{(α, β)} (cos θ) p_k^{(α, β)} (cos φ{symbol})) ≤ c₁ n^{2 max (α, β) + 2} exp (- cn (θ - φ{symbol})²), θ, φ{symbol} ∈ [0, π], n = 1, 2, ... .Specializing to the case of Chebyshev polynomials, α = β = - frac(1, 2), we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L² space.