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Integrability and \(L^1\)-convergence of Rees-Stanojević sums with generalized semi-convex coefficients of non-integral orders. (English) Zbl 1111.42001
Let \(g(x) =\sum \limits _{k=1}^\infty a_k\cos kx\) be a Fourier cosine series and \(g_n(x) =\frac 12 \sum \limits _{k=0}^n \Delta a_k + \sum \limits _{k=1}^n\sum \limits _{j=k}^n (\Delta a_j)\cos kx\). The main result of the paper reads as follows:
If \(a_n\to 0\) as \(n\to \infty \) and \(\sum \limits _{n=1}^\infty n^\alpha \big | \Delta ^{\alpha +1} a_{n-1} + \Delta ^{\alpha +1}a_n\big | <\infty \), where \(\alpha >0\), then \(g_n(x)\) converges in \(L^1\)-metric to \(g(x)\) if and only if \(\Delta a_n \log n = o(1)\) as \(n\to \infty \). This statement is the extension of a similar result presented in K. Kaur and S. S. Bhatia [Int. J. Math. Math. Sci. 30, 645–650 (2002; Zbl 1010.42017)], where it is supposed that \(\alpha \) is an integer. Recall that the noninteger difference \(\Delta ^\alpha a_n\) is defined by the formula \(\Delta ^\alpha a_n = \sum \limits _{m=0}^\infty A_m^{-\alpha -1} a_{n+m}\), where \(A\)’s are binomial coefficient in the expansion \((1-x)^{-\alpha -1}= \sum \limits _{k=0}^\infty A_k^\alpha x^k\).
MSC:
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
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