One-sided exponential approximation in L p-metrics \((0<p\leq 1)\) (Q1100658)

Let f be a real-valued and bounded function in each finite interval and \[ \phi (x)=w(\delta;x,f)\equiv \sup_{u,v\in I_{\delta}(x)}| f(n)-f(v)| \quad (-\infty <x<\infty), \] where \(I_{\delta}(x)=<x- \delta /2,x+\delta /2>.\) The modulus of continuity of f is defined by \[ \tau (\delta;f)_ p=\| \phi \|_ p\equiv \{\int^{\infty}_{- \infty}| \phi (x)| \quad pdx\}^{1/p},\quad 1\leq p<\infty,\quad 0\leq \delta <\infty. \] Let \(\tilde H_{\sigma,p}(f)\) be the set of all pairs (P,Q) of entire functions of exponential type of order \(\sigma\), \(0\leq \sigma <\infty\), such that P(x)\(\geq f(x)\geq Q(x)\forall x\in R\) and P-Q\(\in L\) p. The best one-sided exponential approximation of f is defined by \(\tilde A_{\sigma}(f)_ p=\inf_{(P,Q)\in \tilde H_{\sigma,p}(f)}\| P-Q\|_ p\) if \(\tilde H_{\sigma,p}(f)\neq \emptyset\); \(=\infty\), otherwise. The main results of the paper are embodied in the following two theorems: Theorem I. Let f be a real-valued function such that (1) \(\tau (\delta_ 0,f)_ p<\infty\) for some \(\delta_ 0>0\), \(p\leq 1\). Then \(\tilde A_{\sigma}(f)_ p\leq C(p)\tau (\sigma^{-1};f)_ p\), \(0<\sigma <\infty\). Theorem 2. Let f be a real- valued function belonging to the space M for which the condition (1) holds for \(p\in (0,1>\) and \(\tilde A_{\sigma}(f)_ p<\infty\), then there exists a pair (P *,Q *) of entire functions P \(*\in H\) \(+_{\sigma}(f)\), Q \(*\in H\) \(-_{\sigma}(f)\) such that \(\| P\quad *-Q\quad *\|_ p=\tilde A_{\sigma}(f)_ p,\) where \(H\) \(+_{\sigma}(f)\) is the set of all entire functions \(S\in E_{\sigma}[T\in E_{\sigma}]\) such that S(x)\(\geq f(x)\quad [T(x)\leq f(x)]\) for all x, S-f\(\in M\) and f-T\(\in M\).