Alternating trigonometric polynomials

AbstractTrigonometric polynomials induced by equioscillation with respect to a given periodic function ƒ(t) at appropriately shifted equally spaced nodes are introduced. Two sequences of functionals An(ƒ), Bn(ƒ) (n = 1, 2,…,), corresponding to the specific choice of shift-parameters are defined and their approximation properties are investigated. It is shown that these functionals are closely related to the Fourier coefficients of ƒ(t). It is proved that under some conditions an approximated function is determined by An(ƒ), Bn(ƒ), n = 1, 2,…, uniquely up to an additive constant. It is also shown that the rate at which An(ƒ) and Bn(ƒ) approach zero gives valuable information about the differential properties of ƒ(t)