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On the rate of convergence for the Chebyshev series. (English) Zbl 1031.41010

Let \(U_n(x)\) be the Chebyshev polynomial of the second kind. Let \(f(x)\) be a function of bounded variation on \([-1,1]\) and \(S_n(f;x)\) the \(n\)-th partial sum of the expansion of \(f(x)\) in a Chebyshev series of the second kind: \(\sum _{n=0}^{\infty }a_nU_n(x)\), with \[ a_n={2\over \pi }\int _{-1}^1(1-y^2)^{1/2}f(y){\sin (n+1)\arccos y\over \sin \arccos y}dy,\;(n=0, 1, \dots). \] The paper shows an estimate for the rate of convergence of the sequence \(S_n(f;x)\) to \(f(x)\) in terms of the modulus of continuity of the total variation of \(f(x)\). Specifically the following result is proved:
Theorem. If \(f(x)\) is a continuous function of bounded variation on \([-1,1]\) and \(\omega _{v(f)}(\delta)\) is the modulus of continuity of the total variation \(V_{-1}^t(f)\), then for \(x\in (-1,1)\), \(n\geq 2\) we have \[ |S_n(f;x)-f(x)|\leq {9\over n\sqrt{1-x^2}}\left\{{1\over 1+x}\omega _{v(f)}(1+x)+{1\over 1-x}\omega _{v(f)}(1-x)\right\} \]
\[ +{9\over n\sqrt{1-x^2}}\int _{1/n}^1\left\{{\omega _{v(f)}((1-x)u)\over 1-x}+{\omega _{v(f)}((1+x)u)\over 1+x}\right\}{du\over u^2}. \] When \(V_{-1}^t(f)\) belongs to the class Lip \(\alpha \) (\(\alpha \in (0,1)\)), \[ S_n(f;x)-f(x)=O\left({1\over n^{\alpha }(1-x^2)^{3/2-\alpha }}\right). \] Furthermore, for the Cesàro mean \((c,\lambda)\), \(\lambda \in (0,1)\): \[ \sigma _n^{\lambda }(f;x)={1\over (\lambda)_n\sum _{k=0}^n(\lambda -1)_{n-k}S_k(f;x)},\quad (\beta)_n={\Gamma (\beta +n+1)\over \Gamma (\beta +1)\Gamma (n+1)}, \] we have also \[ \sigma _n^{\lambda }(f;x)-f(x)=O\left({1\over n^{\gamma }(1-x^2)^{3/2-\alpha }}\right) \] where \(\gamma =\min \{\alpha ,1-\lambda \}\).

MSC:

41A25 Rate of convergence, degree of approximation
40A05 Convergence and divergence of series and sequences
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