786 ISSN 1064–5624, Doklady Mathematics, 2007, Vol. 76, No. 2, pp. 786–789. © Pleiades Publishing, Ltd., 2007. Original Russian Text © S.G. Solodky, E.V. Lebedeva, 2007, published in Doklady Akademii Nauk, 2007, Vol. 416, No. 1, pp. 36–39. In this paper, we construct a number of economical discretization schemes, which facilitate improving the efficiency of numerically solving ill-posed problems. Such problems were studied earlier by one of the authors (see, e.g., [1, 2]) in the case where the discrete information about the equation to be solved is given in the form of Fourier coefficients. It turns out that the approaches to constructing economical schemes devel- oped in [1, 2] can also be effectively applied in situa- tions where the discrete information on the integral equation under consideration is given in the form of the values of the kernel and the right-hand side at grid points. (1) Consider the Fredholm integral equation of the first kind (1) under the assumption that x(t), y(t) ∈ L 2 (0, ∞) and a k(t, τ), b(τ) are continuous functions such that, for some constants c k , c b > 0 and any t, τ ∈ [0, ∞), (2) We assume that, instead of the exact right-hand side y(t) of (1), some perturbation y δ (t) ∈ L 2 (0, ∞), where ||y – y δ ≤ δ, is given. The relation Jf (t) = (1 + t) 1/2 f (t) determines an unbounded self-adjoint positive definite operator J on L 2 (0, ∞). Let L 2, s denote the domain of J s endowed Kx t () kt τ , ( ) b τ () x τ ()τ d 0 ∞ ∫ ≡ yt () , t 0, ≥ = kt τ , ( ) c k 1 t + ( ) 1 τ + ( ) [ ] κ – , b τ () c b τ β . ≤ ≤ || L 2 0 ∞ , ( ) with the norm || f || s := 〈 f, f , where 〈 f, g〉 s = 〈J s f, J s g〉 = 1 + t) s f (t)g(t)dt. In what follows, we assume that x ∈ L 2, s and (3) To consider Eq. (1) in the space L 2 (0, ∞), we pass to the new unknown element z(t) ∈ L 2 (0, ∞) such that x(t) := J –s z(t) = (1 + t) –s/2 z(t). Substituting this equality into (1), we obtain the equation (4) The assumption x ∈ L 2, s implies the existence of a con- stant η > 0 for which (5) Suppose that the kernel a(t, τ) = k(t, τ)b(τ)(1 + τ) –s/2 of Eq. (4) satisfies the relation (6) for some reals α, ω > . Example. The equation (τ)τ 4 dτ = y(t) describes the distribution of the size of spherical particles by the scattering method [3]; here, J 1 is the Bessel function of order 1 and x is the unknown distri- bution of the particle size. As is known [4], conditions (2) hold for κ = 3 and β = 4, and the kernel a(t, τ) = (1 + τ) –s/2 , which arises after the change 〉 s 1/2 ( 0 ∞ ∫ κ 1 2 -- , s 2 β 2 κ – 1. + > > Az t () := KJ s – zt () ≡ kt τ , ( ) b τ () 1 τ + ( ) s /2 – z τ ()τ d 0 ∞ ∫ yt () . = x s z L 2 0 ∞ , ( ) η . ≤ = a ij , ( ) t τ , ( ) c 1 t + ( ) α – 1 τ + ( ) ω – , ij , ≤ 01 , = 1 2 -- x 0 ∞ ∫ 2 J 1 t τ , ( ) t τ -------------------- ⎝ ⎠ ⎛ ⎞ 2 2 τ J 1 t τ , ( ) t ----------------------- 2 COMPUTER SCIENCE On Piecewise-Constant Discretization in the Finite Interval Method S. G. Solodky and E. V. Lebedeva Presented by Academician Yu.A. Mitropol’skii October 26, 2006 Received February 27, 2007 DOI: 10.1134/S1064562407050365 Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya ul. 3, Kiev, 01601 Ukraine e-mail: solodky@imath.kiev.ua, djecsa@imath.kiev.ua