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Rate of convergence for the absolutely \((C,a)\) summable Fourier series of functions of bounded variation. (English) Zbl 0952.42003

Let \(f\) be a \(2\pi\)-periodic and Lebesgue integrable function on \([-\pi,\pi]\). Let \(S^\alpha_n(f,x)\) denote the \(n\)th \((C,\alpha)\) mean of the Fourier series of \(f(x)\). It is well known that if \(f\in \text{BV}[-\pi, \pi]\) then its Fourier series is summable \(|C,\alpha|\), \(\alpha>0\), that is to say, \(\sum^\infty_{n=1}|S^\alpha_n- S^\alpha_{n- 1}|< \infty\). Using a result of Riesz if follows that \[ |S^\alpha_n(f, x)-\textstyle{{1\over 2}} \{f(x+ 0)+ f(x- 0)\}|\leq \displaystyle{\sum^\infty_{k= n+1}|S^\alpha_k- S^\alpha_{k-1}|\equiv R^\alpha_n(f, x)}. \] The authors in this paper examine the rate of convergence of \(R^\alpha_n(f, x)\). They prove that if \(f\in \text{BV}[-\pi, \pi]\), then for \(\alpha> 0\), \(n\geq 2\), \[ R^\alpha_n(f, x)\leq {4\alpha\over n\pi} \sum^n_{k=1} \text{Var}^{\pi/k}_0 (\varphi_x),\quad x\in R, \] where \[ \varphi_x(t)= \begin{cases} {1\over 2} f(x+ t)+ f(x- t)- (f(x+ 0)+ f(x- 0)),\quad & t\neq 0,\\ 0,\quad & t= 0.\end{cases}. \]

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
42A10 Trigonometric approximation
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References:

[1] R. Bojanic, An estimate of the rate of convergence for Fourier series of functions of bounded variation, Publications de L’Institut Mathématique. Nouvelle Série, tome 261979, 57-60.; R. Bojanic, An estimate of the rate of convergence for Fourier series of functions of bounded variation, Publications de L’Institut Mathématique. Nouvelle Série, tome 261979, 57-60.
[2] Bojanic, R.; Mazhar, S. M., An estimate of the rate of convergence of Cesàro means of Fourier series of functions of bounded variation, Mathematical Analysis and its Applications. Mathematical Analysis and its Applications, Proceedings of the International Conference on Mathematical Analysis and its Applications (1985), p. 17-22
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