\(k\)-snakes in finite-order interpolational classes of functions (Q1332049)

Let \(C[a, b]\) be the Banach space of all continuous real valued functions on \([a, b]\) with the uniform norm. For two functions \(g_ 0, g_ 1\in C[a, b]\) such that \(g_ 0(x)< g_ 1(x)\) for all \(x\in [a, b]\), the author considers the interpolation class of functions \[ {\mathcal F}(g_ 0, g_ 1)= \{F(x, c)= F(x, c_ 1,\dots, c_ n): g_ 0(x)\leq F(x, c)\leq g_ 1(x)\text{ for all }x\in [a,b]\}. \] A \(k\)-snake is a function on \({\mathcal F}(g_ 0, g_ 1)\) having at least \(n- k+1\) alternation points. Extending his previous results obtained in the case of 2-snakes, the author proves the existence and uniqueness of \(k\)-snakes in finite order interpolation classes and their continuous differentiability dependence on the parameters. It turns out that these \(k\)-snakes are some extremal functions in some interpolation classes of functions.