Interpolation and L1L1-approximation by trigonometric polynomials and blending functions

We present results on interpolation and L1L1-approximation of periodic functions by trigonometric polynomials and trigonometric blending functions. In Section 1, we obtain an error-representation formula for Hermite–Lagrange interpolation by trigonometric polynomials in terms of the differential operator D(2n+1):=D∏k=1n(D2+k2). In Sections 2 and 3, we establish canonical set characterization of the best and best one-sided trigonometric L1L1-approximants under some restrictions. In Section 4, we obtain an error-representation formula for multivariate Hermite–Lagrange transfinite interpolation by trigonometric blending functions that form the kernel of the differential operator Dθ(2m+1)Dη(2n+1). In Section 5, we give explicit constructions of the best trigonometric blending L1L1-approximants to multivariate periodic functions in terms of Hermite–Lagrange transfinite interpolation on canonical sets. Our results on best and best one-sided L1L1-approximation reveal the close relationship between interpolation and best L1L1-approximation (see e.g. Pinkus (1989) [15]). The non-linear problem of best L1L1-approximation becomes a linear interpolation problem on certain convexity functional cones. The interpolation point set of the interpolants that are best L1L1-approximants does not depend on the function to be approximated. For that reason, such an interpolation set is called canonical set of best  L1L1-approximation. In Section  6, we construct one-sided transfinite trigonometric blending interpolants to multivariate periodic functions. Then, we show that the best one-sided trigonometric blending L1L1-approximants to multivariate periodic functions are not transfinite trigonometric blending interpolants on interpolation sets consisting of vertical and horizontal line segments.