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