On the approximation of smooth functions using generalized digital nets

Abstract

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α > 1 in each variable. The approximation error is studied in the worst case setting in the L 2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.