ISSN 0278-6419, Moscow University Computational Mathematics and Cybernetics, 2010, Vol. 34, No. 4, pp. 149–156. © Allerton Press, Inc., 2010. Original Russian Text © D.V. Sorokin, A.S. Krylov, 2010, published in Vestnik Moskovskogo Universiteta. Vychislitel’naya Matematika i Kibernetika, 2010, No. 4, pp. 3–10. 149 1. INTRODUCTION The general scheme of the projection method for solving type I linear equations of Az = u in Hilbert space is based on expanding the solution in a series according to the eigenfunctions of self-adjoint operator A * A [1]. This method is effective for numerically solving a broad class of integral equations. At the same time, modifications of the method that take account of the characteristics of this problem and allow us to improve it are possible for integral equations with kernels of a special type [2]. Let us consider the projection method for the problem of solving an integral equation of the form (1) where the right-hand side is set approximately and J α (x) is a Bessel function of order α. Let us call this the Laguerre projection method of solving the projection problem. Equations of form (1) are used when we need to invert Hankel transform with a finite limit of integra- tion. This transformation is used in problems involving optics [3, 4], geophysics [5, 6], image processing [7], and other fields. One feature of (1) is that the operator A has (from the computational point of view) a multiple eigen- value. This leads to a degradation of the regularized solution when the number of eigenfunctions corre- sponding to this eigenvalue that are used in the projection method is smaller than its multiplicity n α . Table 1 shows the eigenvalues of operator A *A in problem (1) for the values of a = 10 and a = 10.1 at different values of α. The calculation results show that when α = 0, n α = 17 and n α = 18, respectively, for eigenvalue 1 with an accuracy of 10 –14 . When α = 2, n α = 12 and n α = 13, respectively. The eigenfunctions corresponding to these eigenvalues are denoted by ϕ 0 , ϕ 1 , …, . Laguerre functions (x 2 ) = x α (x 2 ), where (x) = (x –α e x /n!) (x n + α e –x ) (n) are eigenfunctions of Hankel transform (a = ∞) of order α and form an orthonormal system in L 2 [0, ∞). At the same time, from a computational point of view, each of these functions focuses on a finite interval [8, 9]. This property is illustrated in Fig. 1, which shows graphs of the dependences of the norms of the Laguerre functions on the length of a segment. It is clear that when a segment is long enough, the norm of the Laguerre functions is close to unity. At the same time, the first Laguerre functions are close to the eigenfunctions of operator A * A in problem (1). Az zx () J α xk ( ) xx d 0 a ∫ uk () , A : L 2 0 a , [ ] L 2 0 a , [ ] , 0 a ∞, < < = = ϕ n α 1 – Ψ n α e x 2 – /2 L n α L n α Laguerre Projection Method for Finite Hankel Transform of Arbitrary Order D. V. Sorokin and A. S. Krylov Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119899 Russia e-mail: dsorokin@cs.msu.ru e-mail: kryl@cs.msu.ru Received March 23, 2010 Abstract—We propose a modification of the projection method for the problem of inverting finite Hankel transform of arbitrary order. In expanding the solution of an integral equation of the first kind, eigenfunctions corresponding to eigenvalues close to the multiple are replaced with Laguerre func- tions. These functions are eigenfunctions of Hankel transform on the half-line. Our test calculations demonstrated the effectiveness of the elaborated method. Keywords: Hankel transform, projection method, Laguerre functions. DOI: 10.3103/S0278641910040011