From high oscillation to rapid approximation IV: accelerating convergence

Modified Fourier expansion is a powerful means for the approximation of nonperiodic smooth functions in a univariate or multivariate setting. In the current paper we consider further enhancement of this approach by two techniques familiar from conventional Fourier analysis: the polynomial subtraction and the hyperbolic cross. We demonstrate that, judiciously subtracting simpler functions dependent on linear combinations of derivatives along boundaries, it is possible to accelerate convergence a great deal and this procedure is considerably more efficient than in the case of conventional Fourier expansion. Moreover, examining the pattern of decay of coefficients in a multivariate setting, we demonstrate that most of them can be disregarded without any ill effect on the quality of approximation.