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