The authors consider the generalized airfoil equation in some weighted Holder-Zygmund spaces with uniform norms. Using a projection method based on the de la Vallée Poussin interpolation, they reproduce the estimates of the L2 case by cutting off the typical extra log m factor which seemed inevitable to have dealing with uniform norm, because of the unboundedness of the Lebesgue constants. The better convergence estimates do not produce a greater computational effort: the proposed numerical procedure leads to solve a simple tridiagonal linear system, the condition number of which tends to a finite limit as the dimension of the system tends to infinity, whatever natural matrix norm is considered. Several numerical tests are given.

In this paper we construct a de la Vallée Poussin approximation process for orthogonal polynomial expansions. Our construction is based on convolution structures which are established by the orthogonal polynomial system. We show that our approach leads to a natural generalization of the de la Vallee Poussin approximation process known from the trigonometric case. Finally we consider Jacobi polynomials and the generalized Chebyshev polynomials expansions as examples.

Recently algebraic polynomials have been considered as wavelets and handled by wavelet techniques. In the unified approach for the construction of polynomial wavelets by Fischer and Prestin, the actual implementation of decomposition, reconstruction and/or compression schemes required at each level the inversion of generalized Grammian matrices, in general not orthogonal. In this context the present paper works out necessary and sufficient conditions for the polynomial wavelets to be orthogonal to each other. Furthermore it shows how these computable characterizations lead to attractive decomposition and reconstruction algorithms based on orthogonal matrices. Finally the special case of Bernstein--Szego weight functions is studied in detail.

The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallee Poussin operator constructed with respect to some Jacobi polynomials.

Using the de la Vallèe Poussin interpolation at the Chebyshev zeros, the authors construct polynomial interpolating wavelets and give the corresponding decomposition and reconstruction algorithms. The involved matrices can be diagonalized by sine and cosine orthogonal matrices. So the algorithms can be realized using fast sine and cosine transforms.

We consider discrete versions of the de la Vallée-Poussin algebraic operator. We give a simple sufficient condition in order that such discrete operators interpolate, and in particular we study the case of the Bernstein-Szego weights. Furthermore we obtain good error estimates in the cases of the sup-norm and L 1-norm, which are critical cases for the classical Lagrange interpolation.