Complexity of approximate realization of functions, satisfying the Lipschitz condition, by schemes in continuous bases (Q920326)

A scheme in the basis \(\{\) x-y,x/2,\(| x|,1\}\) is an arbitrary sequence of functions \(f_ 1,...,f_ L\) such that (i) \(f_ 1(x)=x\) and (ii) if \(f_ j(x)\neq 1\), than there exist indices \(j_ 1,j_ 2<j\) such that either \(f_ j(x)=f_{j_ 1}(x)-f_{j_ 2}(x)\) or \(f_ j(x)=| f_{j_ 1}(x)|\) or \(f_ j(x)=(1/2)f_{j_ 1}(x)\). The number L is called the complexity of the scheme and L(F) denotes the smallest number L for which there exists a scheme of complexity L containing F. For \(f\in C(I)\) with \(I=[a,b]\) and for K a precompact subset of C(I), the complexity L(f,\(\epsilon\)) of the \(\epsilon\)-approximation of f and the Shannon function L(k,\(\epsilon\)) are defined as \(\min \{L(g);\| f- g\| <\epsilon \}\) and max\(\{\) L(f,\(\epsilon\)); \(f\in K\}\), respectively. The purpose of the paper is the proof of the formula \[ L(W(N,M,I),\epsilon)=M| I| /(\epsilon \log M| I| /\epsilon)(1+\Omega (\log \log 1/\epsilon)/\log 1/\epsilon), \] where \(W=W(N,M,I)=\{f\in C(I)\); \(\| f\| <N\), \(\omega\) (f,t)\(\leq Mt\}\) and \(\Omega\) (g(\(\epsilon\))) stands for a function G(\(\epsilon\)) such that \(c_ 1g(\epsilon)<G(\epsilon)<c_ 2g(\epsilon)\) for any \(\epsilon <\epsilon (N,M,I)\) with positive constants \(c_ 1\), \(c_ 2\) independent of \(\epsilon\),M,N,I.