Download e-book for iPad: An Operator Perspective on Signals and Systems by Arthur Frazho, Wisuwat Bhosri

By Arthur Frazho, Wisuwat Bhosri

ISBN-10: 303460291X

ISBN-13: 9783034602914

ISBN-10: 3034602928

ISBN-13: 9783034602921

In this monograph, we mix operator thoughts with kingdom area ways to clear up factorization, spectral estimation, and interpolation difficulties bobbing up on top of things and sign processing. We current either the speculation and algorithms with a few Matlab code to unravel those difficulties. A classical method of spectral factorization difficulties on top of things conception is predicated on Riccati equations bobbing up in linear quadratic keep an eye on concept and Kalman ?ltering. One good thing about this strategy is that it easily ends up in algorithms within the non-degenerate case. however, this procedure doesn't simply generalize to the nonrational case, and it's not regularly obvious the place the Riccati equations are coming from. Operator thought has constructed a few based the right way to end up the life of an answer to a couple of those factorization and spectral estimation difficulties in a really common environment. even if, those innovations are as a rule no longer used to increase computational algorithms. during this monograph, we'll use operator concept with country area how you can derive computational the right way to clear up factorization, sp- tral estimation, and interpolation difficulties. it really is emphasised that our technique is geometric and the algorithms are acquired as a different program of the idea. we'll current tools for spectral factorization. One technique derives al- rithms in response to ?nite sections of a definite Toeplitz matrix. the opposite process makes use of operator conception to increase the Riccati factorization technique. ultimately, we use isometric extension innovations to unravel a few interpolation problems.

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Example text

9). 3. Multiplication Operators 29 Θ H 2 = Θ(eıω ) L2 . So without loss of generality, we can also express Θ as a −ıωn function on the unit circle, that is, Θ(eıω ) = ∞ Θn . Finally, due to this 0 e 2 identification H (E, Y) can be viewed as the subspace of L2 (E, Y) consisting of all ∞ functions of the form 0 e−ıωk Θk . ∞ Recall that H (E, Y) is the Banach space formed by the set of all uniformly bounded, analytic functions in D+ with values in L(E, Y). The H ∞ norm of a function Θ in H ∞ (E, Y), denoted by Θ ∞ , is given by Θ ∞ = sup{ Θ(z) : z ∈ D+ }.

3 Multiplication Operators Let F be a Lebesgue measurable function with values in L(E, Y). 1) where g is a Lebesgue measurable function with values in E. We say that MF is a multiplication operator if (MF g)(eıω ) = F (eıω )g(eıω ) defines an operator, that is, a bounded linear map from L2 (E) into L2 (Y). In this section, we will show that MF defines an operator if and only if F is a function in L∞ (E, Y). Moreover, in this case, MF = F ∞ . Let F be a function in L∞ (E, Y). Then we claim that MF ≤ F ∞ , and thus, MF is a well-defined operator mapping L2 (E) into L2 (Y).

Hence MF ≤ F ∞ . 1 below shows that MF = F ∞ . Notice that MF intertwines the bilateral shift U on L2 (E) with the bilateral shift Z on L2 (Y). 4, we will show that an operator intertwines two bilateral shifts if and only if it is a multiplication operator. We say that a function F is rigid if F is a function in L∞ (E, Y) and F (eıω ) : E → Y is an isometry almost everywhere with respect to the Lebesgue measure. 30 Chapter 2. 1. Let F be a Lebesgue measurable function with values in L(E, Y) over the interval [0, 2π).

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An Operator Perspective on Signals and Systems by Arthur Frazho, Wisuwat Bhosri


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