Application of Chebyshev, and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations. (English) Zbl 0963.65143

Streltsov, I. P.

Comput. Phys. Commun. 126, No. 1-2, 178-181 (2000).

The author proposes to solve Fredholm integral equations of the first and second kind \[ \int_{-1}^1 K(x,y) f(y) dy= q(x), \quad f(x)-\lambda \int_{-1}^1 K(x,y) f(y) dy= q(x), \quad x \in [-1,1] \] by replacing of \(K(x,y), q(x), f(x)\) with their expansions \[ K_n (x,y)=\sum_{k=0}^n \sum_{l=0}^n C_{kl} P_k(x) P_l(y), \quad q_n (x)=\sum_{k=0}^n Q_k P_k (x), \quad f_n (x)=\sum_{k=0}^n e_k P_k(x), \] where \(P_k (x)\) are Chebyshev or Legendre polynomials. Since integral equations of the first kind are ill-posed, the method is not in general correct when applied to such problems.

Reviewer: Mikhail Yu.Kokurin (Yoshkar-Ola)

MSC:

65R20	Numerical methods for integral equations
45B05	Fredholm integral equations
65R30	Numerical methods for ill-posed problems for integral equations

Keywords:

orthogonal polynomials on discrete sets; Chebyshev polynomials; Legendre polynomials; series expansions; ill-posed problems; Fredholm integral equations