tailieunhanh - Handbook of mathematics for engineers and scienteists part 125

Handbook of mathematics for engineers and scienteists part 125. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 836 Integral Equations As was noted above the eigenfunctions corresponding to distinct characteristic values are orthogonal. Hence the sequence of eigenfunctions of a symmetric kernel can be made orthonormal. In what follows we assume that the sequence of eigenfunctions of a symmetric kernel is orthonormal. We also assume that the characteristic values are always numbered in the increasing order of their absolute values. Thus if Ai X2 . Xn . is the sequence of characteristic values of a symmetric kernel and if a sequence of eigenfunctions i 2 . n . corresponds to the sequence so that pn x - An J K x t pn t dt 0 then b I pi x pj x dx 1 for j Ja ej 0 for i j v 7 and Ai A2 An . If there are infinitely many characteristic values then it follows from the fourth Fredholm theorem that their only accumulation point is the point at infinity and hence An to as n to. The set of all characteristic values and the corresponding normalized eigenfunctions of a symmetric kernel is called the system of characteristic values and eigenfunctions of the kernel. The system of eigenfunctions is said to be incomplete if there exists a nonzero square integrable function that is orthogonal to all functions of the system. Otherwise the system of eigenfunctions is said to be complete. . Bilinear series. Assume that a kernel K x t admits an expansion in a uniformly convergent series with respect to the orthonormal system of its eigenfunctions K x t ak x pk t k 1 for all x in the case of a continuous kernel or for almost all x in the case of a square integrable kernel. We have ak x K x t pk t dt k x J a k and hence K x t V t . k k 1 . Linear Integral Equations of the Second Kind with Constant Limits of Integration 837 Conversely if the series f Q t k 1 fc is uniformly convergent then formula holds. The following assertion holds the bilinear series converges in mean-square to .

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