System approximation from later time samples on a generalized temporal grid

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Let {fn}n=1∞ be a Schauder basis for L2([0,1]). We consider a function u(x,t) which can be represented as(1)u(x,t)=∑n=1∞anfn(x)gn(t). In our manuscript, our primary focus is to investigate under what conditions the initial condition function f(x)=u(x,0) can be reasonably approximated from the provided samples {u(x0,tk)}k=1N. Our research encompasses a diverse spectrum of partial differential equations (PDEs) that describe the Initial Value Problems, particularly emphasizing the initial condition functions. Assuming that the initial condition function belongs to a predefined function class in Sobolev space, we develop conditions required to achieve the desired level of accuracy N−r in reconstructing the solution. Effectively controlling errors of recovered coefficients of the approximated initial condition function, to contain a linear growth term, we enhanced the efficiency of the recovery process compared to previously known reconstruction algorithms.