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Proceedings of the Symposium on Progress in Data Processing and Analysis

Quantization means, that values a^ are assigned to a function f:
Q = f - { a k} k -
These values may be determined locally, i. e., in connection with the
digitized points x^, or maybe obtained globally. This assignment of values
can be connected now with the development of the function with the help of
corresponding function systems. Thereby the numbers a^ are the coefficients
of a corresponding expansion.
In Table 1 these processes and function systems are compared.
Practically the following assignements of values are of great importance:
- function values in the digitized points (scanned values),
- interpolation of function values within a neighbourhood
(convolution of rectangle function),
- FOURIER-coefficients of the function (harmonic analysis).
3. Functional analytical methods
The above described functionals can interpreted as elements of an. abstract
space - the so-called dual space.
Thus, a bilinear form (f,x) can be defined, which is linear in both components
and which assigns a number to a pair of elements f and x. Two essential
examples are shown in Figure 2.
A HILBERT-space corresponds to its dual one, and it can be shown, for every
linear functional there exists an element out of the space itself, which
creates this functional by the scalar product. These facts are important,
because assignment of numbers to an analogue signal can be understood as
discrete realization of the signal itself, i. e. it leads to an equivalence
of discrete and analogue representation.
With integral representations continuous and discrete signals or realizations
in form of scalar products can be described. For continuous representations
this integral is a true one, whereas with discrete realizations it decomposes
into a series. This is analogue to the equivalence of the HILBERT-spaces of
the square integral functions L 2 and the square summable sequences l 2 (see
also Fig. 3).