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.

The Cauchy-Schwarz inequality and the

approximation property (3.

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.

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.

Since we do not require an

approximation property of these basis functions, the construction is only based on a reference element.

We show that the homogeneous

approximation property and the comparison theorem hold for arbitrary coherent frames.

h] are chosen to satisfy a certain compatibility condition known as discrete inf-sup condition together with a certain

approximation property, then it is well-known [15, 16], under a certain shape-regularity of [T.

The remaining terms can be treated using the Neumann

approximation property from Theorem 4.