Communications in Mathematics and Applications Volume 3 (2012), Number 3, pp. 253–259 © RGN Publications http://www.rgnpublications.com Approximated Solutions to Operator Equations based on the Frame Bounds H. Jamali and A. Askari Hemmat Abstract. We want to find the solution of the problem Lu = f , based on knowledge of frames. Where L : H → H is a boundedly invertible and symmetric operator on a separable Hilbert space H. Inverting the operator can be complicated if the dimension of H is large. Another option is to use an algorithm to obtain approximations of the solution. We will organize an algorithm in order to find approximated solution of the problem depends on the knowledge of some frame bounds and the guaranteed speed of convergence also depends on them. 1. Introduction The analysis of numerical schemes for operator equations is a field of enormous current interest. In this work we present an algorithm in order to approximate the solution of the operator equation Lu = f , (1.1) where L : H → H is a boundedly invertible and symmetric operator on a separable Hilbert space. As typical example we think of linear differential or integral equations in variational form. In [1, 5, 6], some iterative adaptive methods for solving this system has been developed. First natural steps were to use multiresolution spaces spanned by wavelets (or correspondingly scaling functions) as test and trial spaces for Galerkin methods. Usually, the operator under consideration is defined on a bounded domain Ω ⊂ R d or on a closed manifold, and therefore the construction of a wavelet basis with specific properties on this domain or on the manifold is needed. Although there exist by now several construction methods such as e.g., [7, 8], none of them seems to be fully satisfying in the sense that some serious drawbacks such as stability problems cannot be avoided. One way out could be to use a fictitious domain method [9], however, then the compressibility of the problem might be 2010 Mathematics Subject Classification. 39B42, 65K15. Key words and phrases. Operator equation; Hilbert space; Frame; Approximated solution.