On an analogue of the Lizorkin-Nikol'skij inequality for an infinite interval (Q1110017)

Let \(L^ r_{p,\alpha}\) be the space of functions f(t) with finite functional \(\| t^{\alpha}f^{(r)}(t)\|_{L_ p(1,+\infty)}\), \(r\in N\), \(\alpha\in {\mathbb{R}}\), \(1\leq p\leq +\infty\). It is known that each function \(f\in L^ r_{p,\alpha}\) stabilizes in a definite sense to some polynomial \(P(t)=\sum^{r-1}_{s=0}a_ st^ s\) as \(t\to +\infty\). The inequality \[ | f(x)| \leq c(\sum^{k-1}_{\nu =0}| f^{(i_{\nu})}(1)| +\sum^{l-1}_{\mu =0}a_{j_{\mu}}+\| t^{\alpha}f^{(r)}(t)\|_{L_ p(1,+\infty)}) \] is proved, where \(k+l=r\), and \(i_ 0,...,i_{k-1}\) and \(j_ 0,...,j_{l-1}\) are certain admissible collections of indices.