Evolutionary systems representations based on later time samples and applications to PDEs

Let { f_n}^∞_n=₁ be a basis for L₂([0, 1]) and {g_n}^∞_n=₁ be a system of functions of controlled decay on [0, ∞). Considering a function u(x, t) that can be the represented in the form ∞ u(x, t) = X a_n f_n(x)g_n(t), n=1 where a_n ∈ R, x ∈ [0, 1] and t ∈ [0, ∞), we investigate whether the function f (x) = u(x, 0) can be approximated, in a reasonable sense, by using data u(x₀, t₁), u(x₀, t₂),..., u(x₀, t_N). A mathematical framework and efficient computational schemes are developed to determine approximate solutions for various classes of partial differential equations via sampled data by first establishing a near-best approximation of f.