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Pointwise Convergence of Eigenfunction Expansions Associated with Ordinary Differential Operators

By Hendrik S.V. de Snoo


With an ordinary differential expression L = ∑nk=0pkDk on an open interval I⊂R is associated a selfadjoint operator H in a Hilbert space, possibly beyond H=L2(ι). The set DH∩H only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of ι for functions in D(H)∩H. In case H is a semibounded selfadjoint operator in H=L2(ι), a similar result is proved for functions in D[H], which is the set of all f∈H for which there exists a sequence fn∈D(H) such that fn→f in H and (H(fn − fm), fn − fm) → 0 as n, m → ∞.

Year: 1983
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