Abstract
Based on our decomposition of stochastic processes and our asymptotic representations of Fourier cosine coefficients, we deduce an asymptotic formula of approximation errors of hyperbolic cross truncations for bivariate stochastic Fourier cosine series. Moreover we propose a kind of Fourier cosine expansions with polynomials factors such that the corresponding Fourier cosine coefficients decay very fast. Although our research is in the setting of stochastic processes, our results are also new for deterministic functions.
Highlights
For approximations of multivariate functions by algebraic/ trigonometric polynomials on full grids, the approximation rate deteriorates rapidly as the dimension d increases [1, 2]; this is just so-called “dimension curse” problem
Based on the above decomposition, we show that Fourier cosine coefficients of a bivariate stochastic process ξ on [0, 1]2 can be approximated asymptotically by
Based on our decomposition of stochastic processes, we propose Fourier cosine expansions with polynomial factors (see (111)) whose hyperbolic cross truncations are a combination of stochastic algebraic polynomials and stochastic cosine polynomials
Summary
For approximations of multivariate functions by algebraic/ trigonometric polynomials on full grids, the approximation rate deteriorates rapidly as the dimension d increases [1, 2]; this is just so-called “dimension curse” problem. In 2010, Shen and Wang [1] studied the hyperbolic cross Jacobi polynomial approximation for functions on the unit cube and gave various formulas on error estimates. We will deeply study the hyperbolic cross approximation in Fourier cosine analyses and give a precise asymptotic formula of approximation error of hyperbolic cross truncation. For hyperbolic cross approximation, we give the following precise result: if ξ is a stochastic process on [0, 1]2 with smoothness index l ≥ 2, its hyperbolic cross truncations SN(h)(ξ) (see (16)) of stochastic Fourier cosine series of ξ satisfy the following asymptotic formula:. We see that hyperbolic cross approximation with polynomial factors can reconstruct the stochastic process on [0, 1]2 by using the least Fourier cosine coefficients.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
