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Zbl 0963.65143
Streltsov, I.P.
Application of Chebyshev, and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations. (English)
[J] Comput. Phys. Commun. 126, No.1-2, 178-181 (2000). ISSN 0010-4655

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.
[Mikhail Yu.Kokurin (Yoshkar-Ola)]

MSC 2000:: *65R20 Integral equations (numerical methods)
45B05 Fredholm integral equations
65R30 Improperly posed problems (integral equations, numerical methods)

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

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Zentralblatt MATH Berlin [Germany]