Quadrature formulas for Fourier coefficients

Error bounds of a quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients for analytic functions

“…where ℓ i,ν are the fundamental polynomials of the Hermite interpolation and Ωn,s(z) = n ν=1 (z − xν) 2s . If we choose xν to be the zeros of the Chebyshev polynomial of the first kind, i. e., xν = ξν , after multiplying by (2) with ω(t)Tn(t), where ω(t) = 1/ √ 1 − t 2 , and integrating in t over (−1, 1), we get a contour integral representation of the remainder term in (1).…”

“…which is based on the divided differences of f ′ at the zeros of the Chebyshev polynomial Tn. For more details on this subject see [1], [2], [3], [11], [13].…”

Quadrature formula for computed tomography

“…Formulas of type (3.1) have been investigated in [8]. Here we state one of the theorems derived in [8], which applies to our case.…”

Error estimations of Turán formulas with Gori-Micchelli and generalized Chebyshev weight functions

“…, n − 1, is connected with the Gauss-Turán quadrature formula (1). It is well known that the zeros of a polynomial π n are the nodes τ ν of the generalized Gauss-Turán quadrature formula (1). The weight coefficients A i,ν from (1) are determined through interpolation and they are not all positive in general.…”

“…ADP of the formula (1) is 2(s+1)n−1. Numerically stable methods for constructing nodes τ ν and coefficients A i,ν in Gauss-Turán quadrature formulas with multiple nodes (1), and their generalizations, are treated in several papers; see [3], [4], [12], [13], [15], [22]; see also the book [2]. For more details on Gauss-Turán quadrature formulas and corresponding s-orthogonal polynomials, including answers on the questions of their existence and uniqueness, see [5], [14], [6], [18], [19].…”