1
$\begingroup$

If $\left( p_n \right)_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x_j, w_j \right)$ are the quadrature points and weights for the respective Gaussian quadrature rule, we can easily prove that $$ f_N (x) : \, = \sum\limits_{n=0}^{N-1} \sum\limits_{j=1}^{N} p_n (x_j) f(x_j) w_j p_n (x) \, ,$$ is the interpolation polynomial of degree $N-1$ for $f$ at the quadrature points $x_1, \ldots x_N$.

My question: While I could prove it, I couldn't find a reference for this proof in any textbook. Could you help with that?

This is cross-posted from this post in MSE.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Here, three possible references for the formula:

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Computer Science and Applied Mathematics. Academic Press, New York, 1984 (see p.88)

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press, New York, 2003 (see section 8.3.2 for the case of Chebyshev Polynomials)

J. Shen, T. Tang, L-L. Wang, Spectral methods. Algorithms, analysis and applications. Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011 (see Thm 3.9 p.64)

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thanks! I didn't find it in Davis and Rabinowitz, because I only have the 1967 edition. Could you write in what part is it, so I can look it up? $\endgroup$
    – Amir Sagiv
    Commented Feb 18, 2018 at 9:30
  • $\begingroup$ I think the formula appears in section 2.5.6 : Product integration rules. But, in the 3rd reference, the book by Shen, Tang, Wang, the formula is maybe better presented. $\endgroup$
    – user111
    Commented Feb 18, 2018 at 11:28
  • $\begingroup$ Thanks, but we don't have it in our library... $\endgroup$
    – Amir Sagiv
    Commented Feb 18, 2018 at 11:41

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .