Abstract
This chapter describes the expansion theorems. The Arzela-Ascoli theorem states that every equibounded and equicontinuous sequence of functions {fn (x)} defined on an interval [a, b] has a uniformly convergent subsequence on that interval. Every eigenfunction of K corresponding to an eigenvalue λ is a finite linear combination of the functions φk such that λk = λ. It follows that every eigenfunction f corresponding to an eigenvalue λ of Km is also an eigenfunction corresponding to the same eigenvalue of K. The theory of ordinary differential equations, the principal expansion theorem states that every function v ∈ C2 [a, b] that satisfies a second-order differential equation and homogeneous boundary conditions can be expanded in a uniformly and absolutely convergent Fourier series.
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