Polynomial Expansions of Solutions of Cauchy Problems That Involve One Space Variable

AbstractPolynomial sets and polynomial expansion theorems are developed for solutions of Cauchy problems of the form ∂nu(x,t)/∂tn=±Dmu(x,t), ∂ju(x,t)/∂tj|t=0=φj(x),j=0,1,…,n−1. Ifm>n, then the data functions must be entire of growth ρ=s/(s−1) wheres=m/nifndividesmand of growth ρ=(s+1)/sifm=ns+rwhere 1≤r≤n−1. Ifn≥m, thenR-analyticity of the φj(x) will suffice