On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions

“…This gives the edge to the Zernike basis in most applications. However, the Logan-Shepp basis lends itself well to solving the Poisson equation through the Dual Reciprocity Method as discussed above, and has been applied to PDEs by Atkinson, Hansen and Chien [7,49,4,5,8].…”

“…Atkinson, Hansen and Chien have successfully used Logan-Shepp polynomials to solve the nonlinear Poisson equation (reformulated as an integral equation) and other partial differential equations [7,49,4,5,8]. One strength of their work is that they prove exponentially fast convergence in solving PDEs.…”

New cubature formulae and hyperinterpolation in three variables

“…[16,18,11,5]). The hyperinterpolation has been used effectively in several cases: originally for the sphere [16,18], and more recently for the square [4,5], the disk [11], and the cube [6]. We will use our new cubature formulae to construct a hyperinterpolation operator of three variables for the Chebyshev weight function on the cube.…”

Convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type

“…In the following Lemma, proceeding with the techniques used in [15] and [25], we derive the error bound for the hyperinterpolation operator in the infinity norm. …”