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Approximation with Bernstein-Szegö polynomials. (English) Zbl 1119.41005

The authors consider the approximation properties of modified orthogonal expansions. They consider Szegő-Bernstein weights \[ w(x)= {1\over \sqrt{1- x^2}\rho(x)},\quad x\in (-1,1), \] where \(\rho\) is a polynomial positive on \([-1, 1]\). The advantage of these weights is the explicit formula for their orthogonal polynomials. Let \(\{p_n\}^\infty_{n=0}\) denote the corresponding orthonormal polynomials, so that \[ \int^1_{-1} p_n p_m w= \delta_{mn}. \] For functions \(f:(-1, 1)\to\mathbb{R}\) for which \(fw\) is integrable, we may form the Fourier coefficients \[ \widehat f(k)= \int^1_{-1} fp_k,\quad k\geq 0, \] and modified partial sums of the orthogonal expansion, \[ A_N f= \sum^N_{k=0} a_{N,k}\widehat f(k) p_k. \] Assuming that \(\sup_{N,k}\,|a_{N,k}|< \infty\) and for some \(m\), \[ \sup_{N\geq m}\, \sum^{N-m}_{k=0} |a_{N,k+ m}- a_{N, k}|< \infty, \] the authors show that uniform boundedness of the operators \(\{A_N\}^\infty_{N= 1}\) does not depend on which polynomial \(\rho\) appears in \(w\). Thus one may reduce the general case to that of the Chebyshev weight, and hence to clasical Fourier series. They deduce that for general \(\rho\), a variety of choices of \(\{a_{N,k}\}\) lead to boundedness, for example, those arising from the Fejér, Rogosinski, de la Vallée-Poussin, and Fejér-Korovkin kernels.

MSC:

41A10 Approximation by polynomials
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