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From: chernoff@dirac.berkeley.edu
Newsgroups: sci.math,sci.physics,sci.electronics,sci.philosophy.tech
Subject: Re: distributed transformations (request)
Message-ID: <8825@agate.BERKELEY.EDU>
Date: 16 Apr 88 23:51:33 GMT
References: <8694@cisunx.UUCP>
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Organization: Math Dept., UC Berkeley
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Article 3510 of sci.math:

   <<I am interested in finding as many and varied examples as possible
   <<of functions or transformations with either or both of the following
   <<properties
   <<
   <<(1) equidistribution: each output element depends on the whole input
   <<(in some relevant sense of input).  An example is the Fourier 
   <<transform which computes every point in the transform function
   <<on the basis of the entire input function.
   <<
   <<(2) Redundancy: the input information is recoverable not just from the
   <<whole ouput but from proper portions of it (in the extreme case, from
   <<arbitrary portions).  An intuitive example of this phenomenon is the
   <<hologram, where the whole encoded scene is recoverable from any portion
   <<of the photographic plate.
   <<
   <<All suggestions are welcome (no matter how obscure, unusual or bizarre).
   <<
   <<Thanks
   <<Tim van Gelder
   <<vangelde@unx.cis.pittsburgh.edu

------------------
An interesting example is the (unilateral) Laplace transform
which is both 'equidistributed' and 'redundant'.  On one hand the
Laplace transform F(s) of a given function f(t) depends on the 
function f over the entire interval (0,inf). On the other hand,
F(s) is an analytic function of s, so knowing it in any non-
empty open subset of its domain determines it and hence the
original function f(t).
Analogously, the "z-transform" of a sequence {a_n} is an analytic
function A(z) = sum_0^inf A_n z^n, (provided this series has a
positive radius of convergence), and so knowing it on any small
open subset of its region of convergence determines it and hence the
original coefficient sequence.
Or, use the Cauchy integral formulas to determine the a_n from the
values of F(z) on any closed curve enclosing 0 .

Other examples occur in partial differential equations; for
example, consider the Dirichlet or Neumann boundary value
problems for the Laplace equation.

It would be interesting to have examples of these phenomena
which do NOT involve analyticity in some sense.

*************************************************************************
* Paul R. Chernoff                       chernoff@cartan.berkeley.edu   *
* Department of Mathematics              ucbvax!cartan!chernoff         *
* University of California                                              *
* Berkeley, CA  94720                                                   *
*************************************************************************
*************************************************************************
* Paul R. Chernoff                       chernoff@cartan.berkeley.edu   *
* Department of Mathematics              ucbvax!cartan!chernoff         *
* University of California                                              *