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On linear methods of summations of a Fourier series. (English) Zbl 0607.40003

The authors prove the following result: Theorem. For the triangular matrix \((\alpha _{n,k})\) of positive elements such that \(\alpha _{n,k}\to 1\) as \(n\to \infty\), if \(\sum ^{n}_{k=0}\frac{(k+1)(n- k)}{n+1}| \Delta ^ 2\alpha _{n,k}| \leq A\) and \(\phi (t)=o(1/\log t^{-1})\) as \(t\to 0\), then \[ \sum ^{n}_{k=0}\frac{\alpha _{n,k}}{(n-k+1)\log (n-k+1)}\leq A, \] if and only if the Fourier series of f(t) is summable (\(\Lambda)\) to f(x) at \(t=x\). Here \(\Delta \alpha _{n,k}=\alpha _{n,k}-2\alpha _{n,k+1}+\alpha _{n,k+2}\) and A is an absolute constant.

MSC:

40C05 Matrix methods for summability
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