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

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