Integral Equations

Replacing x by y abd vice versa, we obtainso thatNow we haveIn the same manner, we obtainand, generally,∫1bφ n (y) = K(y, x)φ n (x)dx, (229)λ n aω n (y) = 1 λ nφ n (y).K 2 (x, y) =K 3 (x, y) =K m (x, y) =which is bilinear series for kernel K m (x, y).For kernel K(x, y), the bilinear series isK(x, y) =∞∑n=1∞∑n=1∞∑n=1∞∑n=1φ n (x)φ n (y). (230)λ 2 nφ n (x)φ n (y),λ 3 nφ n (x)φ n (y). (231)λ m nφ n (x)φ n (y)λ n. (232)This series may diverge in the sense of C-norm (i.e., uniformly); however it alwyas convergesin L 2 -norm.11.2 Hilbert–Schmidt theorem for integral operatorsConsider a Hilbert–Schmidt integral operator A with a symmetric kernel K(x, y). In this case,the following conditions are assumed to be satisfied:i Aφ(x) =iiiii∫ b ∫ baa∫ baK(x, y)φ(y)dy,|K(x, y)| 2 dxdy ≤ ∞, (233)K(x, y) = K(y, x).According to Theorem 18, A is a completely continuous selfadjoint operator in the space L 2 [a, b]and we can apply the Hilbert–Schmidt theorem 24 to prove the following statement.45