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ON SUPERPOSITIONS OF CONTINUOUS FUNCTIONS A. A. Agrachev UDC 517.5
 

Summary: ON SUPERPOSITIONS OF CONTINUOUS FUNCTIONS
A. A. Agrachev UDC 517.5
We show that if r is an arbitrary countable set of continuous functions of n variables, then
there exists a continuous, and even infinitely smooth, function r 1..... x n) such that * (~,,
.... ~) * ~[~ (~,(~,)..... s~ (~,))] for any function ~ from ff and arbitrary continuous functions g
and fi' depending on a single variable.
1~ In k-valued logics Slupecki's criterion(see [1], for example), which gives a necessary and suffi-
cient condition for the completeness of systems containing all functions of a single variable, is widely
known.
In the generalization of this criterion to a countably valued logic (see [2]) it was found that if the func-
tion r ..... Xn), together with all functions of a single argument, forms a complete system, then an arbi-
trary function r 1.... , Xn) can be obtained in the form of a superposition relative to ~o of order not higher
than the second, wherein only functions depending on a single variable are used, i.e., in the form
(xl .... , x.) = go (qD(el [~ (/11 (xl) ..... /~,~ (x~))], 9 9 9 g~ [q) (/~ (x~) ..... I~ (x.))])).
Analogous problems for continuous functions are also of no small interest inasmuch as A. N. Kol-
mogorov [3] showed that an arbitrary function r 1..... Xn), continuous on the unit n-dimensional cube, can
be written in the form
~_~2rt+l ~n
where gm ~ C [0, l], i,~z ~ C I0, t] for l = 1, 9.., n; m = 1.... ,2n+ 1, i.e., as a completely specific super-
position of continuous functions of a single argument and addition (in Kolmogorov's construction the func-

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics