Integral Equations

A²(x algebraic analytic function approximation asymptotic B²(y basic interval belongs boundary conditions C₁ C₂ Carleman class L₂ Consequently consider corresponding deduced denotes determinant dx dy eigenfunctions eigenvalues fact finite follows formula Fourier coefficients Fredholm equation Fredholm integral equation given equation given function f(x Green's function Hence Hilbert-Schmidt theorem homogeneous equation hypothesis infinite number infinite series iterated kernels K²(x kernel K(x L₂-function L₂-kernel linear combination linear differential equation linearly independent Math membrane method Moreover non-homogeneous non-linear integral equation non-trivial solutions obtain ON-system orthogonal PG-kernel Pn(x polynomials positive previous section problem prove resolvent kernel respectively Riesz-Fischer theorem right-hand side satisfies condition Schwarz inequality second kind shows singular suitable symmetric kernel transformation Tricomi uniformly convergent values vanish almost everywhere Volterra Equations Volterra integral equation zero λ₁ λ²π² λη λι λπ