A note on the representation of continuous functions by linear superpositions

We consider the problem of the representation of real continuous functions by linear superpositions ∑i=1kgi∘pi with continuous gigi and pipi. This problem has been considered by many authors. But complete and at the same time explicit and practical solutions to the problem were given only for the case k=2k=2. For k>2k>2, a fairly practical sufficient condition for the representation can be found in Sternfeld (1978) [17] and Sproston and Strauss (1992) [16]. In this short note, we give a necessary condition of such a kind for the representability of continuous functions.