| This thesis mainly analyzes the reasons of the error of the "Lanczos-Fox-Parker" proposition on the comparison of Chebyshev and Legendre expansion coefficient decay rate.The Chebyshev and Legendre expansion coefficients have been extensively applied in the numerical solution of differential equations, and the convergence of the series expansion is linked to the decay rates of the expansion coefficients. Generally, the faster the expansion coefficients, the faster the convergence of the series is. So it is meaningful to study the coefficients of the series expansion.In the first chapter, we describe the research background and current research of orthogonal polynomial expansion, especially the Chebyshev and Legendre expansion.In the second chapter, we describe the comparison results of Chebyshev and Legendre coefficients of “Lanczos-Fox-Parker” proposition, and analyze the reason of the error of“Lanczos-Fox-Parker” proposition. For functions whose singularity is outside or at the endpoints of the expansion interval, we compare the Chebyshev coefficients with those calculated by “Lanczos-Fox-Parker” proposition, and perform numerical experiments on the entire functions, and the function whose singularity is inside of the expansion interval.Similarly, we also analyze the Legendre expansion coefficients,and demonstrate the Chebyshev coefficients and Legendre coefficients of the “Lanczos-Fox-Parker” proposition are only correct for a few of the entire function,while for most of the entire function,function whose singularity are outside, inside and at the endpoints of the expansion interval,the “Lanczos-Fox-Parker” proposition is incorrect. Then, we analyze “Lanczos-Fox-Parker”proposition of the function with two singular points, and it is found that the proposition is not correct. The above conclusions are also verified by numerical experiments. |
