Approximated Solutions to Operator Equations based on the Frame Bounds

H. Jamali, A. Askari Hemmat


We want to find the solution of the problem ${\L} u = f$, based on knowledge of frames. Where ${\L} : H \to H$ 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.


Operator equation; Hilbert space; Frame; Approximated solution

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A. Askari Hemmat and H. Jamali, Adaptive Galerkin frame methods for solving operator equations, U.P.B. Sci. Bull., Serries A 73(2) (2011), 129–138.

P.G. Casazza, Modern tools for Eyl-Heisenberg (Gabor) frame theory, Ddv. in Imag. and Electeron. Physics 115(2001), 1–127.

P.G. Casazza, The art of frame theory, Taiwaness J. Math. 4 (2000), 129–201.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003.

A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. of Comp. 233(70) (2001), 27–75.

A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelets methods II-beyond the elliptic case, Found. of Comp. Math. 2 (2002), 203–245.

W. Dahmen and R. Schneider, A composite wavelet bases for operator equations, Math. Comp. 68 (1999), 1533–1567.

W. Dahmen and R. Schneider, Wavelets on manifolds I. Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999), 184–230.

M. Mommer, Fictitious domain Lagrange multiplier approach: Smoothness analysis, Report 230, IGPM, RWTH Aachen, 2003.



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