Sharp (with respect to the order of the highest exponent) estimate of the derivative of a power quasipolynomial in \(L_ 2[0,1]\) (Q802091)

A Markov type estimation for quasi-polynomials with bounded mean density of exponents is obtained, i.e., for every sequence of functions \(f_ N(x)=a_ 0+\sum_{\gamma \in \Gamma_ N}\sum^{{\mathcal K}^ N_{\gamma}-1}_{k=0}a_{\gamma,K}x^{\gamma}(\ell n x)^ K\), with the \(\Gamma_ N\subset [1/2+\epsilon,N]\), \(\epsilon >0\), \(N>1/2+\epsilon\), \({\mathcal K}^ N_{\gamma}\in {\mathbb{N}}\) and such that for some number \(a>0\) there exists \(m\in {\mathbb{N}}\cap [a,\infty [\) for which \(\max_{x\in {\mathbb{R}}}\sum_{\gamma \in \Gamma_ N\cap]x,x+a]}{\mathcal K}^ N_{\gamma}\leq m\), the inequality \(\| f'\!_ N\|_{L_ 2[0,1]}\leq CN^{2m/a}\| f_ N\|_{L_ 2[0,1]}\) (where C does not depend on a, m, \(\epsilon)\) is true.