Math. Nachr. 279, No. 11, 1147 – 1158 (2006) / DOI 10.1002/mana.200510414 A Hausdorff-like moment problem and the inversion of the Laplace trans- form Nguyen Dung ∗1 , Nguyen Vu Huy ∗∗2 , Pham Hoang Quan ∗∗∗2 , and Dang Duc Trong †2 1 Institute of Applied Mechanics, Vietnamese Academy of Science and Technology, 291 Dien Bien Phu Str., Dist. 3, Hochim- inh City, Vietnam 2 Hochiminh City National University, 227 Nguyen Van Cu, Q5, Hochiminh City, Vietnam Received 30 August 2005, accepted 28 November 2005 Published online 7 July 2006 Key words Ill-posed problem, inversion of the Laplace transform, Hausdorff moment problem, Muntz poly- nomial, polynomial approximation MSC (2000) Primary: 65R30; Secondary: 41A10 Dedicated to Professor Phan Dinh Dieu We consider the problem of finding u ∈ L 2 (I ), I = (0, 1), satisfying Z I u(x)x α k dx = μ k , where k =0, 1, 2,... , (α k ) is a sequence of distinct real numbers greater than -1/2, and μ =(μ kl ) is a given bounded sequence of real numbers. This is an ill-posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. c  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction This paper deals with two problems. The first one is that of finding u in L 2 (I ), I = (0, 1), satisfying  I u(x)x α k dx = μ k (1.1) where k =0, 1, 2,... , (α k ) is a sequence of distinct real numbers such that α k > - 1 2 for all k =0 , 1 , 2 ,..., and (μ k ) is a given bounded sequence of real numbers. The second problem is that of approximating u 0 in L 2 (0, +∞) such that  ∞ 0 u 0 (x)e -β k x dx = μ 0 k , k =0 , 1 , 2 ,..., (1.2) ∗ Corresponding author: e-mail: nguyenschiller@gmail.com, Phone: (84.8) 9300944, Fax: (84.8) 9308300 ∗∗ e-mail: tquan@pmail.vnn.vn, Phone: (84.8) 8350098, Fax: (84.8) 8350096 ∗∗∗ e-mail: namolienhoasanh@hcm.vnn.vn, Phone: (84.8) 8350098, Fax: (84.8) 8350096 † e-mail: ddtrong@mathdept.hcmuns.edu.vn, Phone: (84.8) 8350098, Fax: (84.8) 8350096 c  2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim