Generalized Laguerre Approximation In Multiple Dimensions And Their Applications On Unbounded Domains

The spectral methods are based on smooth orthogonal polynomials to approximate the solution of mathematical physics problems.Its advantage is high precision,which is called exponential order convergence.When the solution of the problem is sufficiently smooth,the spectral methods can use smaller number of degrees of freedom to get better approximation results.In this paper,we mainly discuss the efficient spectral method of high-dimensional high-order problems on unbounded domains.Firstly,we construct a two dimensional generalized Laguerre function system by using one dimensional generalized Laguerre function,and establish corresponding orthogonal approximation and interpolation ap-proximation theory;Secondly,we extend the above results in the case of arbitrary di-mensions.These approximation results provide a powerful tool for the spectral methods of high-dimensional high-order problems.Finally,we take the two dimensional case as an example,we construct the corresponding spectral scheme and give specific nu-merical analysis,and the effectiveness of the proposed method is verified by numerical experiments.