TY - JOUR

T1 - Pointwise Convergence of Eigenfunction Expansions Associated with Ordinary Differential Operators

AU - Snoo, Hendrik S.V. de

N1 - Relation: http://www.rug.nl/fmns-research/bernoulli/index
Rights: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science

PY - 1983

Y1 - 1983

N2 - 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 → ∞.

AB - 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 → ∞.

M3 - Article

SN - 0022-247X

VL - 92

SP - 172

EP - 179

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -