Constrained mock-Chebyshev least squares quadrature

Product integration rules by the constrained mock-Chebyshev least squares operator

“…and let the function f be known only at the points of X n . The main idea of the constrained mock-Chebyshev least squares method [9,13] is to construct an interpolant of f on a proper subset of X n , formed by m +1 nodes, chosen as "closest" to Chebyshev-Lobatto nodes, and use the remaining n − m points of X n to improve the accuracy of approximation by a process of simultaneous regression of degree p ≥ 0. To be more precise, let m = π n 2 , and denote by X C L m the set of Chebyshev-Lobatto nodes of order m + 1…”

“…, ẑm+1 ] T the Lagrange multipliers vector. In defining V and C in (11) the assumption is that the nodes ξ i have been reordered so that ξ i = ξ i , i = 0, . .…”

“…In this setting, we propose a product integration rule, based on the approximation of f known on a set of equispaced nodes, by using the constrained mock-Chebyshev least squares linear operator Pr,n ( f ) [9,12]. Such operator has been recently used for deriving an efficient quadrature rule [10,11] based on equidistant points, for integrals of the type (1), with K (x, y) ≡ 1. As we will show, the product formula we introduce here is convergent in suitable subspaces of C([−1, 1]), providing in these cases estimates of the quadrature error.…”

A product integration rule on equispaced nodes for highly oscillating integrals

“…In order to adopt the values of f at equidistant nodes, a widely used technique is based on composite quadrature rules of "lower" degree, such as Filon's type rules, whose degree of approximation cannot be improved over the saturation class of the approximation process. Recently, in [3] starting from the knowledge of f at a finite set of equally spaced nodes, an approach has been introduced to efficiently compute weighted integrals, but not advisable in the case of oscillating integrands. For a short review on the main techniques the interested reader can consult [1].…”

A product integration rule on equispaced nodes for highly oscillating integrals

“…In order to adopt the values of f at equidistant nodes, a widely used technique is based on composite quadrature rules of "lower" degree, such as Filón type rules, whose degree of approximation cannot be improved over the saturation class of the approximation process. Recently, in [3] starting from the knowledge of f at a finite set of equally spaced nodes, an approach has been introduced to efficiently compute weighted integrals, but not advisable in the case of oscillating integrands. For a short review on the main techniques the interested reader can consult [1].…”