[sci.electronics] distributed transformations

vangelde@cisunx.UUCP (Timothy J Van) (04/15/88)

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

chernoff@dirac.berkeley.edu (04/17/88)

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                                              *

doug@eris (Doug Merritt) (04/18/88)

In article <8694@cisunx.UUCP> vangelde@cisunx.UUCP (Timothy J Van) writes:
>I am interested in finding as many and varied examples as possible
>of functions or transformations with either or both of the following [...]
>
>(1) equidistribution: each output element depends on the whole input

Try Walsh functions (like Fourier but with square waves instead of sine).

>(2) Redundancy: the input information is recoverable not just from the

The theory of error correcting codes applies. What I've read about
it is based on the idea of N-space spheres/circles. Each encoding
represents a point in the space. Each point is surrounded by a circle
of constant radius. If the circles overlap, a bit error is regarded
as a transformation that takes you into the next circle over, and you
can't correct it. If they don't overlap, an error moves you over a
little but it's still unambiguous which circle you're in. Sorry
for being vague (and any errors), I'm not an expert on this.

Laplace transforms in general would probably interest you, too.
How about matrix inversion?

	Doug Merritt		doug@mica.berkeley.edu (ucbvax!mica!doug)
			or	ucbvax!unisoft!certes!doug