The behavior of the Lagrange polynomial Lm.w; f /, based on the zeros
of the orthogonal polynomials, is studied in some weighted Besov spaces Bp
r;q .u/. It is
proved that Lm.w/ is a uniformly bounded map under suitable conditions on the weight
functions and the parameters p, r , and q
The mapping properties of the Canchy singular integral operator with constant\ud
coefficients are studied in couples of spaces equipped with weighted uniform\ud
norms. Recently weighted Besov type spaces got more and more interest in\ud
approximation theory and, in particular, in the numerical analysis of polynomial\ud
approximation methods for Cauchy singular integral equations on an interval. In\ud
a scale of pairs of weighted Besov spaces the authors state the boundedness and\ud
the invertibility of the Canchy singular integral operator. Such result was not\ud
expected for a long time and it will affect further investigations essentially.\ud
The technique of the paper is based on properties of the de la Vall'èe Poussin\ud
operator constructed with respect to some Jacobi polynomials
We introduce and study the sequence of bivariate Generalized Bernstein operators {Bm,s}m,s, m, s ∈ N, Bm,s = I − (I − Bm) s , B i m = Bm(B i−1 m), where Bm is the bivariate Bernstein operator. These operators generalize the ones introduced and studied independently in the univariate case by Mastroianni and Occorsio [Rend. Accad. Sci. Fis. Mat. Napoli 44 (4) (1977), 151-169] and by Micchelli [J. Approx. Theory 8 (1973), 1-18] (see also Felbecker [Manuscripta Math. 29 (1979), 229-246]). As well as in the one-dimesional case, for m fixed the sequence {Bm,s(f)}s can be successfully employed in order to approximate "very smooth" functions f by reusing the same data points
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