Application of Chebyshev, and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations
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.
scientific article; zbMATH DE number 1437197