On a Bojanić-Stanojević type inequality and its applications. (English) Zbl 0970.42004

Summary: An extension of the Bojanić-Stanojević type inequality [R. Bojanić and C. V. Stanojević, Trans. Am. Math. Soc. 269, 677-683 (1982; Zbl 0493.42011)] is made by considering the \(r\)-th derivate of the Dirichlet kernel \(D_k^{(r)}\) instead of \(D_k\). Namely, the following inequality is proved: \[ \Bigl\|\sum_{k=1}^n \alpha_k D_K^{(r)} (x)\Bigr\|_1\leq M_pn^{r+1}\Bigl({1\over n}\sum^n_{k=1}|\alpha_k|^p\Bigr)^{1/p} \] where \(\|\cdot\|_1\) is the \(L^1\)-norm, \(\{\alpha_k\}\) is a sequence of real numbers, \(1<p\leq 2, r=0,1,2,\dots\) and \(M_p\) is an absolute constant dependent only on \(p\). As an application of this inequality, it is shown that the class \(\mathcal F_{pr}\) is a subclass of \(\mathcal B\mathcal V\cap\mathcal C_r\), where \(\mathcal F_{pr}\) is the extension of the Fomin’s class, \(\mathcal C_r\) is the extension of the Garrett-Stanojević class [Ž. Tomovski, Approximation Theory Appl. 16, No. 1, 46-51 (2000; Zbl 0963.42003)] and \(\mathcal B\mathcal V\) is the class of all null sequences of bounded variation.


42A20 Convergence and absolute convergence of Fourier and trigonometric series
26D10 Inequalities involving derivatives and differential and integral operators
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