Bounded operator

Summary

In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, |Lx|_Y \leq M |x|_X. The smallest such M is called the operator norm of L and denoted by |L|. A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called "bounded" then this usually means that its f(X) is a bounde

Official source

https://en.wikipedia.org/wiki/Bounded_operator

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Exponential decay and Fredholm properties in second-order quasilinear elliptic systems

Hicham Georges Gebran, Charles Stuart

We consider second-order quasilinear elliptic systems on un-bounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C-1-Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabies approach to proving exponential decay to zero at infinity of solutions. (C) 2010 Elsevier Inc. All rights reserved.

2010

Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

Anaïs Laure Marie-Thérèse Badoual, Julien René Fageot, Michaël Unser

This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a trade off between fidelity to the data and smoothness conditions via a quadratic regularization associated with a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two. We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method.

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