Quadrature rules of Gaussian type for trigonometric polynomials with preassigned nodes.

In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the interval E = [ a , a + 2 π) , a ∈ R , these quadrature rules have the following form ∫ E t (x) w (x) d x = ∑ i = 1 2 k a i t (y i) + ∑ i = 1 2 (n + γ) A i t (x i) , t ∈ T 2 (n + γ) + k − 1 , where the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance, γ ∈ { 0 , 1 / 2 } and T n = { cos ⁡ k x , sin ⁡ k x | k = 0 , 1 , ... , n } , n ∈ N. Also, for γ = 1 / 2 , i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see [1,13]) for trigonometric polynomials with even number of fixed nodes. Let n = (n 1 , n 2 , ... , n r) , r ∈ N , be a multi-index and let W = (w 1 , w 2 , ... , w r) be a system of weight functions on the interval E = [ a , a + 2 π) , a ∈ R. The optimal set of quadrature rules with respect to (W , n) , with even number of fixed nodes, have the form ∫ E f (x) w m (x) d x ≈ ∑ i = 1 2 k a m , i f (y i) + ∑ i = 1 2 | n | + 1 A m , i f (x i) , m = 1 , 2 , ... r , where | n | = n 1 + n 2 + ⋯ + n r and the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance. For r = 1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes. For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples. [ABSTRACT FROM AUTHOR]