(a) the

approximation property if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional linear map L:Y [right arrow] Y such that L(y) - y [omega] O for every y [omega] K;

From the

approximation property (3.5) with r = q and t = l + 1, we easily obtain

namely, we have in mind the

approximation property, the Bohr property which served as definition (existence of the almost periods) and the Bochner property (the compactness of the family of translates).

ABSTRACT: A non-empty subset S of a valued field K is said to have the optimal

approximation property if every element in the field K has a closest approximation in S.

But because of disregarding the universal

approximation property, SNP and FLNs are not very popular in the applications.

From the

approximation property of the L[degrees]-projection, we have

This latter equation holds if and only if X has the compact

approximation property (see [GS], Cor.

And finally, an MLP network with sufficient hidden neurons can satisfy the universal

approximation property [29].

Looking at recent results in the area of ergodic theory (the mathematical study of dynamical systems with an invariant measure) concerning the complexity of the problem of classification of ergodic measure preserving transformations up to conjugacy, the structure of the outer automorphism group of a countable measure preserving equivalence relation, ergodic theoretic characterizations with the Haagerup

approximation property, and cocycle superrigidity, the author of this monograph realized that these apparently diverse results can all be understood within a unified framework.

In the case of Whittaker operator, for functions with positive values, the newly obtained nonlinear sampling operator has essentially a better

approximation property than its linear counterpart.

al l991) do not have the universal

approximation property, but Ridged-Polynomial network (Y.Shin, et.

There are many versions of the

approximation property for a Banach space [chi] but all have a common theme: there exists a net {T[alpha]: [chi] [right arrow] [chi]}[alpha][element of]A of finite rank maps converging to I in an appropriate topology.