ON ENTIRE FUNCTIONS AND A FOURIER INTEGRAL PROBLEM
Abstract
Some previous work of Akutowicz on the following problem is given: that (x),( L2), is null outside some finite interval, what can be said about , if it is known that its Fourier transform, , satisfies (x) = a(x), where a(x) is some fixed function In some cases, the the question yields easily if we consider a(x) 2, which is readily seen to be continuable as a function of exponential type. Specifically, if Lp, (p > 2), the problem is tractable in the sense that it always has a solution (subject to simple conditions on a(x), and the totality of solutions K(x) can be displayed, not directly, but through their Fourier transforms, K(x). If, on the other hand, Lp, (1 < p < 2), existence conditions which involve only a(x) do not appear, and we find it necessary to augment the hypotheses. Even with augmented hypotheses, the conclusions are substantially weaker than in the case p > 2. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 05, 1961
- Accession Number
- AD0259568
Entities
People
- J.a. Sheehan
Organizations
- Massachusetts Institute of Technology