Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa

Quasi orthogonal Jacobi polynomials and best one-sided L1 approximation to step functions

“…Many interlacing properties of the zeros, x (α,β) r, j , varying the parameters r , α and β, are known (see, for example [1,7,12]). Listed below are those we will use in this paper:…”

“…Section 4 is devoted to study the question whether there exists a positive quadrature formula having the point θ as a prescribed node. We use the results in [1] where the authors give a complete solution of this problem. Finally, in Section 5 we provide explicit expressions for the polynomials in P − n (G θ ) and describe the cases when we have uniqueness.…”

One-sided weighted integral approximation of characteristic functions of intervals by polynomials on a closed interval

“…Let us formulate the main result of [10] using the introduced terms. Positive quadrature formulas are used (see Theorem B) for the lower estimation of values (2.1) of the best one-sided (weighted integral) approximation to a bounded function f ∈ L by polynomials.…”

“…There is a large number of papers devoted to such formulas (see [9,10,12] and references therein). These formulas are rich in applications.…”

“…These formulas are rich in applications. The most convenient for our purposes formulas were obtained in [9,10]. To find upper bounds, we use polynomials constructed according to methods going back to Markov and Stieltjes (see the beginning of this section); later, these methods were developed in [1, Lemma 1; 8, Sect.…”