From: Robin Chapman Subject: Re: Paley-Wiener Theory (not theorem) Date: Tue, 11 Jan 2000 13:34:07 GMT Newsgroups: sci.math Summary: [missing] In article <387b1143_4@news1.prserv.net>, "R. Joseph Lyons" wrote: > > Has anyone heard of Paley-Wiener Theory? > > Can you give me a reference to some book/article > and/or a description of what Paley-Wiener Theory is? > > I am familiar with the Paley-Wiener Theorem (or Criterion) which can > be used to determine which loss functions alpha(f) > alpha(f) = Log|H(f)|, H(f) some transfer function > can be associated with a causal system. > > I am looking for info on Paley-Wiener Theory; > I am not looking for info on the Paley-Wiener Theorem, described above. > (unless info on the Theorem is somehow "included" in PW Theory; > in which case I'm more interested in the general theory than the particular > criterion). This does not look much like the Paley-Wiener theorems of which I am aware. I must admit though, that I am not familiar with jargon such as "loss functions", "transfer function" and "causal system". The Paley-Wiener theorems I am aware of deal with functions f holomorphic on strips of the form {z: a < Im z < b} and such that sup_y integral |f(x+iy)|^2 dx < infinity. (That is the L^2 norms of the functions obtained by restricting f to horizontal lines are finite and uniformly bounded on the strip). One of their results states that if f is holomorphic on the half plane {z: Im z > 0} and sup_{y>0} integral |f(x+iy)|^2 dx < infinity then there is an L^2 function F on R with f(x+iy) -> F(y) in L^2 norm as x -> 0. Also the Fourier transform of F vanishes for negative argument. The converse is also true. One can find this and other Paley-Weiner type results in Katznelson's "An Introduction to Harmonic Analysis" published by Dover. -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: "David C. Ullrich" Subject: Re: Paley-Wiener Theory (not theorem) Date: Tue, 11 Jan 2000 10:53:02 -0600 Newsgroups: sci.math Robin Chapman wrote: [most of previous article was quoted --djr] > This does not look much like the Paley-Wiener theorems of which I am > aware. I must admit though, that I am not familiar with jargon such as > "loss functions", "transfer function" and "causal system". I'm not familiar with that stuff either but I tend to suspect there's a connection. Say a "filter" is something you convolve with - if S=S(t) is a "signal" (defined on R or on Z) and F is a filter then the result of applying the filter to the signal is the convolution S*F. If we assume the signal is arriving bit by bit and we want to ouput the filtered signal bit by bit as the original arrives then the fact that we can't look into the future shows we must have F(t) = 0 (t >0, or probably rather t < 0 because of the minus sign in the convolution...) Now if we assume that F is in L^2, just because then we can do stuff, and we consider the Fourier Transform, for various reasons, we see that F^ is an H^2 function. If we think of the convolver F as instead a multiplier on the "frequency" side we might wonder about the magnitude of F^. And the functions that can be |F^| with F^ in H^2 are exactly the functions with integrable log... (I saw more or less the above in Dym&McKean a few years ago - being unfamilar with the jargon I don't know that what he's talking about has anything to do with what I wrote above, but it seems likely.) [remainder of previous article was quoted, now deleted --djr] ============================================================================== From: "Daniel H. Luecking" Subject: Re: Paley-Wiener Theory (not theorem) Date: Mon, 17 Jan 2000 13:33:00 -0600 Newsgroups: sci.math.research On Mon, 17 Jan 2000, Robin Chapman wrote: > In article <3880a6db_1@news1.prserv.net>, > "R. Joseph Lyons" wrote: > > > > Has anyone heard of Paley-Wiener Theory? > > > > Can you give me a reference to some book/article > > and/or a description of what Paley-Wiener Theory is? "Fourier Transforms in the Complex Domain" by Raymond E. A. C. Paley and N. Weiner, Amer. Math. Soc, Colloquium Puplications, vol. 19, 1934. > > I am familiar with the Paley-Wiener Theorem (or Criterion) which can > > be used to determine which loss functions alpha(f) > > alpha(f) = Log|H(f)|, H(f) some transfer function > > can be associated with a causal system. This sounds vaguely like "prediction theory", in which a discrete (or continuous) time stationary gaussian process is associated with a family of subspaces of L^2 of some measure on the unit circle (or real line). Various properties of the stochastic series are associated with function theoretic properties of the contiuous part of that measure. As my area is complex function theory, I have seen one end of such stuff (the function theoretic properties), but am not too familiar with the other end (the stochastic series) or the translation between them. > This does not look much like the Paley-Wiener theorems of which I am > aware. I must admit though, that I am not familiar with jargon such as > "loss functions", "transfer function" and "causal system". Nor am I, but the function log h, where h is continuous part of the measure mentioned above, is crucial in prediction theory. Its integrability is (I believe) somehow associated with independence of "past" and "future" of the stochastic process. It is relatex to the function theoretic result that a positive function in L^2 whose log is integrable is the absolute value of the boundary values of some function analytic in the unit disk (or upper half-plane). "Time Series", by Norbert Weiner, M.I.T. Press, might have something on the subject. -- Dan Luecking Dept. of Mathematical Sciences luecking@comp.uark.edu University of Arkansas http://comp.uark.edu/~luecking/ Fayetteville, AR 72101 ============================================================================== From: "David C. Ullrich" Subject: Re: Paley-Wiener Theory (not theorem) Date: Tue, 18 Jan 2000 11:58:33 -0600 Newsgroups: sci.math.research "Daniel H. Luecking" wrote: > On Mon, 17 Jan 2000, Robin Chapman wrote: > > > In article <3880a6db_1@news1.prserv.net>, > > "R. Joseph Lyons" wrote: > > > > > > Has anyone heard of Paley-Wiener Theory? > > > > > > Can you give me a reference to some book/article > > > and/or a description of what Paley-Wiener Theory is? > > "Fourier Transforms in the Complex Domain" by Raymond E. A. C. Paley and > N. Weiner, Amer. Math. Soc, Colloquium Puplications, vol. 19, 1934. A classic. Possibly the reader should be advised that "l.i.m." means "limit in mean", that a "closed" orthonormal set is a _complete_ set, etc. > > > > I am familiar with the Paley-Wiener Theorem (or Criterion) which can > > > be used to determine which loss functions alpha(f) > > > alpha(f) = Log|H(f)|, H(f) some transfer function > > > can be associated with a causal system. > > This sounds vaguely like "prediction theory", in which a discrete (or > continuous) time stationary gaussian process is associated with a family > of subspaces of L^2 of some measure on the unit circle (or real line). > Various properties of the stochastic series are associated with function > theoretic properties of the contiuous part of that measure. > > As my area is complex function theory, I have seen one end of such stuff > (the function theoretic properties), but am not too familiar with the > other end (the stochastic series) or the translation between them. > > > This does not look much like the Paley-Wiener theorems of which I am > > aware. I must admit though, that I am not familiar with jargon such as > > "loss functions", "transfer function" and "causal system". > > Nor am I, but the function log h, where h is continuous part of the > measure mentioned above, is crucial in prediction theory. Its > integrability is (I believe) somehow associated with independence of > "past" and "future" of the stochastic process. It is relatex to the > function theoretic result that a positive function in L^2 whose log is > integrable is the absolute value of the boundary values of some > function analytic in the unit disk (or upper half-plane). There's a neat book by Dym&McKean, "Fourier something something", that says something about this. (In terminology readers here will understand, but written so non-mathematicians will have a chance as well.) The integrability of log h is what makes the thing an H^2 function, as you say. The Paley-Weiner theorem says that's the same thing as an L^2 function f with f^(t) = 0 for t < 0 . Or t > 0, depending on where the minus signs are. And the fact that f^(t) = 0 for t > 0 is what says that your filter can't look into the future. > "Time Series", by Norbert Weiner, M.I.T. Press, might have something on > the subject. > > -- > Dan Luecking Dept. of Mathematical Sciences > luecking@comp.uark.edu University of Arkansas > http://comp.uark.edu/~luecking/ Fayetteville, AR 72101