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On some transforms of trigonometric series. (English) Zbl 0901.65100
Let $$(a_n)_{n=1}^\infty$$ be a sequence of real or complex numbers. For the generalized forward difference $$\Delta_r^k(a_n)$$ defined by $$\Delta_r(a_n)=\Delta_r^1(a_n)=a_{n+1}-ra_n$$, $$\Delta_r^{k+1}(a_n)=\Delta_r(\Delta_r^k(a_n))$$, where $$r$$ is a real or complex number, different from $$1$$, and $$k=1,2,\ldots$$, a method for accelerating the convergence of the trigonometric series $$\sum_{n=1}^\infty a_n\cos(\alpha n+\beta)x$$ was presented by G. A. Sorokin [Sov. Math. 28, No.11, 41-48 (1984); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1984, No. 11(270), 34-40 (1984; Zbl 0571.40001)]. Generalizations of such a method for numerical and power series were given by I. Ž. Milovanović, M. A. Kovačević, and S. D. Cvejić [Sov. Math. 32, No. 4, 120-122 (1988); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1988, No. 4(311), 82-84 (1988; Zbl 0702.40002)] and I. Ž. Milovanović, M. A. Kovačević, S. D. Cvejić, and J. Klippert [J. Nat. Sci. Math. 29, No. 1, 1-9 (1989; Zbl 0723.40001)], respectively. The authors used the linear operator $$L_{r_1\cdots r_k}(a_n)$$ instead of $$\Delta_r^k(a_n)$$, which is defined by $$L_{r_1}(a_n)=a_{n+1}-r_1a_n$$, $$L_{r_1\cdots r_kr_{k+1}}(a_n)=L_{r_1\cdots r_k}(a_{n+1})-r_{k+1}L_{r_1\cdots r_k}(a_n)$$, where $$(r_\nu)_{\nu=1}^\infty$$ is a given sequence.
The authors of the paper under review use the same linear operator in order to obtain the corresponding transformations for the cosine and sine trigonometric series. They also give sufficient conditions for the faster convergence of the transformed series than the original one.
##### MSC:
 65T40 Numerical methods for trigonometric approximation and interpolation 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 65B10 Numerical summation of series
##### Keywords:
trigonometric series; acceleration of convergence
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