Chapter 7  Approximation Theory

 

Approximation by Orthogonal Polynomials and Fourier Series

 

 

Continuous Fourier transform and inverse transform:

,    ，

， 

,    。

Discrete Fourier transform and inverse transform:

,  ,

   ，

  ，

   。

Continuous Fourier transform and inverse transform (in complex form):

,    ，

Discrete Fourier transform and inverse transform (in complex form):

,  ,

，

.

There are two ways to compute the above . One is by matrix multiplication with number of multiplication of , the other is by FFT with number of multiplication of .

Continuous Legendre transform and inverse transform:

,    ，

where  is the k-th Legendre polynomial.

，   .

Discrete Legendre-Gauss transform and inverse transform:

,  ,

，

where  are roots of , and

Discrete Legendre-Gauss-Radau transform and inverse transform:

,  ,

，

where  are roots of , and 

Discrete Legendre-Gauss-Lobatto transform and inverse transform:

,  ,

where  are , roots of , and 1. 

   

Continuous Chebyshev transform and inverse transform:

,    ，

where  is the k-th Chebyshev polynomial.

, and

，   .

Discrete Chebyshev-Gauss transform and inverse transform:

,  ,

where  and

Discrete Chebyshev-Gauss-Radau transform and inverse transform:

,  ,

where  and 

Discrete Chebyshev-Gauss-Lobatto transform and inverse transform:

,  ,

where  and 

   

 

Aliasing

   The relation of  and  in complex Fourier series is

       ,    .

It shows that the k-th mode of the trigonometric interpolant of  depends not only on the k-th mode of , but also on all the modes of  which “alias” the k-th one on the discrete grid. The  -th frequency aliases the k-th frequency on the grid; they are indistinguishable at the nodes since , where .

   Similarly, for Chebyshev polynomial expansion,  and  in Chebyshev-Gauss-Lobatto transform are related as:

 

      ,    .

That means the  -th frequency aliases the k-th frequency on the grid; they are indistinguishable at the nodes since , where .

 

 

References:

【1】         R. L. Burden and J. D. Faires, Numerical Analysis, PWS, Boston, 1993.

【2】         B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1996.

【3】         C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.