The eigenvalue behavior of certain convolution equations

Abstract

Introduction.In a series of papers [3], [4], [6], we studied the relationship between two closed subspaces of L\ -oe, oe): the subspace 3>T of all fEL2 supported in \t\ < T/2 and the subspace 3$a of all fEL2 whose Fourier transforms are supported in |w| < Í2/2.We showed that several questions about 2#T and ^a could be answered in terms of the eigenvalues of the operator BaDfBa, where Ba and DT are the projections onto 3ßa and irrespectively;this operator may be written as a finite convolution.Apart from this application, interpretable as describing the way in which the energy of a function of L2 can be distributed over time and over frequency, the behavior of these eigenvalues is interesting because it differs markedly from that established by H. Widom[7] for the class of finite convolutions with L1 kernels whose Fourier transforms have an absolute maximum at the origin.By a change of variable, the eigenvalues of BaDTBü may be seen to depend on the parameter c = ííT/2ir, rather than on Í2 and T separately; we may write their equation explicitly as Xn(cW?(t) = -P* sinf~x) tf>ix) dx> n = 0,1,2, ..., * J -c/2 t -X and we suppose that X0 ^ Xx ^ • • •.For any fixed c, the X"(c), ra = 0,1, • • •, form a positive sequence bounded away from 1 and approaching 0 at a rate in n greater than (ce/ra)2" [D.Slepian, unpublished].For any fixed ra, the eigenvalue X"(c) approaches 1 exponentially in c [2].In [4] we proved, however, that X[cj+1(c) is bounded away from 1 independently of c, and interpreted this to imply that the set of functions in 38^ whose energy is concentrated in |i| < T/2 has, in a well-defined sense, approximate dimension bounded by [S2T/2ir](1).We also showed that X^^c) is bounded away from 0 independently of c.The analogous questions for the case where the intervals |i| < T/2 and | w| < 12/2 are replaced by more general sets T' and Q' have not been studied.Indeed, most of the methods developed to deal with rSa are not applicable to ^a, and very little is known about it.Here we give another, simpler, proof that X|cj+1(c) andX[cj_i(c) are bounded away from 1 and 0 respectively,

Locations

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