#### Title

Localized nonlinear functional equations and two sampling problems in signal processing

#### Abbreviated Journal Title

Adv. Comput. Math.

#### Keywords

Nonlinear functional equation; Strict monotonicity; Inverse-closedness; Banach algebra; Infinite matrix; Instantaneous companding; Average; sampling; Signal with finite rate of innovation; Shift-invariant space; SHIFT-INVARIANT SPACES; FINITE SECTION METHOD; OFF-DIAGONAL DECAY; INFINITE MATRICES; WIENERS LEMMA; PSEUDODIFFERENTIAL-OPERATORS; RECONSTRUCTING SIGNALS; BANACH-ALGEBRAS; INNOVATION; FRAMES; Mathematics, Applied

#### Abstract

Let 1 a parts per thousand currency sign p a parts per thousand currency sign a. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to a"" (p) and f has continuous bounded gradient in an inverse-closed subalgebra of a"not sign (a""(2)), the Banach algebra of all bounded linear operators on the Hilbert space a"" (2). We introduce strict monotonicity property for functions f on Banach spaces a"" (p) so that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in a"" (p) . We show that the Van-Cittert iteration converges in a"" (p) with exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.

#### Journal Title

Advances in Computational Mathematics

#### Volume

40

#### Issue/Number

2

#### Publication Date

1-1-2014

#### Document Type

Article

#### Language

English

#### First Page

415

#### Last Page

458

#### WOS Identifier

#### ISSN

1019-7168

#### Recommended Citation

"Localized nonlinear functional equations and two sampling problems in signal processing" (2014). *Faculty Bibliography 2010s*. 6151.

https://stars.library.ucf.edu/facultybib2010/6151

## Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu