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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.)
##### Keywords:
conjugate Cesàro mean; Fourier cosine series
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