ABSTRACT
We present results on existence, uniqueness, and explicit characterization of Birkhoff type quadrature formulae on equidistant nodes having maximal trigonometric degree of precision. Here the data means lacunary data; i.e., the derivatives taken at a given node are not necessarily consecutive [7]. The corresponding trigonometric interpolation problem in the case of k-periodic data was first proposed and solved in [11]. However, the necessary and sufficient conditions for the existence and uniqueness of the trigonometric interpolant were found in a simple form only for k = 1 [1] (equal multiplicities). In the case of trigonometric interpolation with two-periodic multiplicities data some special cases have been explicitly solved. Details can be found in [10]. In view of this we found interesting to find explicitly quadrature formulae of maximal trigonometric degree of precision based on two-periodic, Birkhoff type data, to study their existence and uniqueness by making use of a direct method, without using any prior knowledge of the corresponding interpolants that even may not exist [2].
