Sharp Ul'yanov inequalities for generalized Liouville-Weyl derivatives
The main focus of the article is sharp \(\left(p,q\right)\)-inequalities between fractional moduli of smoothness. Inequalities of this type are called Ul'yanov-type inequalities. The author considers these inequalities for the generalized Liouville-Weyl derivatives defined in the following way. Let the Fourier series of \(f\in L_1\left({\mathbb T}\right)\) be given by \[ f\left(x\right)\sim\sum_{\nu\in{\mathbb Z}}\hat f_\nu e^{i\nu x}, \] where \(\hat f_\nu\) are the Fourier coefficients of \(f\). The generalized Liouville-Weyl derivative is defined by (provided the right-hand side is a Fourier series of an integrable function) \[ D^\lambda f\left(x\right)\sim\sum_{\nu\in{\mathbb Z}}\lambda_\nu\hat f_\nu e^{i\nu x}, \] where \(\lambda_\nu=\lambda\left(\left| \nu\right| \right)\), for some non-decreasing function \(\lambda:\,\left[0,\infty\right)\to\left[0,\infty\right)\). The author cites several Ul'yanov-type results for these derivatives valid when \(\lambda\) satisfies certain properties and notices at the same time that those results do not cover limiting cases (when \(p=1\), or \(q=\infty\)). The present article proves more general analogues of the aforementioned results, in particular, considering monotonicity requirement replaced by the general monotonicity. Here, a sequence \(\left\{\lambda_n\right\}_{n=1}^\infty\) is said to be general monotone if the relation \[ \sum_{k=n}^{2n}\left| \lambda_k-\lambda_{k+1}\right| \leq C\left| \lambda_n\right| \] holds for all \(n\in{\mathbb N}\) with the constant \(C\) independent of \(n\). Furthermore, the limiting cases are also covered. Let us cite one of the results obtained. Theorem. Let \(f\in L_p\), \(1=p<q<\infty\), \(\theta=1-1/q\). Let the sequence \(\left\{\lambda_n\right\}_{n=1}^\infty\) be general monotone. Then, for any \(\alpha>0\) and \(0<\varepsilon\leq\min\left(\rho,\theta\right)\), \[ \omega_\alpha\left(D^\lambda f,\frac{1}{2^n}\right)_q\lesssim \left(\sum_{m=n}^\infty\left(2^{m\left(\theta+\rho\right)}\Lambda_{2^m} \omega_{\alpha+\rho+\theta-\varepsilon}\left(f,\frac{1}{2^m}\right)_1\right)^q\right)^{1/q}, \] where \[ \Lambda_{2^m}:=2^{-m\varepsilon/2}\max_{1\leq k\leq m}\frac{\left| \lambda_{2^k}\right|}{2^{k\left(p-\varepsilon/2\right)}}. \] Other interesting results obtained in the article are estimates for the best approximation of the generalized Liouville-Weyl derivatives in \(L_q\) by trigonometric polynomials via the best approximation in \(L_p\) of the function itself. Let us cite one of the results. Theorem. Let \(f\in L_p\), \(1<p<q\leq\infty\), \(\theta=1/p-1/q\). Let \(\left\{\lambda_n\right\}_{n=1}^\infty\) be general monotone. Then, for any \(\alpha>0\), \[ E_{2^n}\left(D^\lambda f\right)_q\lesssim\left(\sum_{m=n}^\infty\left(2^{m\theta}\left| \lambda_{2^m}\right| E_{2^m}\left(f\right)_p\right)^{q_1}\right)^{1/q_1}, \] where \(q_1=q\) if \(q<\infty\) and \(q_1=1\) if \(q=\infty.\) The sharpness of the inequalities is also shown. There are several interesting remarks throughout the article. In particular, it is interesting that Theorem 2.3 not only generalizes, but also sharpens one of the results by \textit{S. Tikhonov and W. Trebels} [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 1, 205--224 (2011; Zbl 1213.26010)]. The article should be interesting for specialists in approximation theory and harmonic analysis.
scientific article; zbMATH DE number 6837907
scientific article