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By Arthur Frazho, Wisuwat Bhosri

During this monograph, we mix operator options with nation house how to resolve factorization, spectral estimation, and interpolation difficulties coming up up to the mark 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 up to speed idea relies on Riccati equations coming up in linear quadratic regulate thought and Kalman ?ltering. One benefit of this technique is that it comfortably results in algorithms within the non-degenerate case. nevertheless, this strategy doesn't simply generalize to the nonrational case, and it's not consistently obvious the place the Riccati equations are coming from. Operator concept has constructed a few stylish how you can end up the life of an answer to a few of those factorization and spectral estimation difficulties in a truly normal atmosphere. even though, those options are quite often now not used to strengthen computational algorithms. during this monograph, we are going to use operator conception with nation area tips on how to derive computational tips on how to resolve factorization, sp- tral estimation, and interpolation difficulties. it truly is emphasised that our strategy is geometric and the algorithms are acquired as a different software of the speculation. we are going to current equipment for spectral factorization. One process derives al- rithms according to ?nite sections of a undeniable Toeplitz matrix. the opposite technique makes use of operator conception to boost the Riccati factorization process. ultimately, we use isometric extension concepts to unravel a few interpolation difficulties.

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2. In other words, TΘ = TΨ W = TΨ TΩ . Therefore Θ(z) = Ψ(z)Ω for all z in D+ . 1. 2. As before, let M be an invariant subspace for the unilateral shift SY on 2+ (Y). Let Φ be any isometry mapping a space E into 2+ (Y) such that the range of Φ equals M SY M. The proof of the Beurling-Lax-Halmos Theorem shows that Θ(z) = (FY+ Φ)(z) is an inner function in H ∞ (E, Y) satisfying M = TΘ 2+ (Y). 7). 1, we obtain the following H 2 version of the BeurlingLax-Halmos theorem. 3. Let S be a unilateral shift on H 2 (Y), then M is an invariant subspace for S if and only if M = ΘH 2 (E) where Θ is an inner function in H ∞ (E, Y).

Let SE be the unilateral shift on L2+ (E) and UE the bilateral shift on L2 (Y). Then an operator T is in I(SE , UE ) if and only if T = MF |L2+ (E) where MF is a multiplication operator and F is a function in L∞ (E, Y). In this case, T = F ∞ . Finally, T is an isometry if and only if F is rigid. 5 Toeplitz Operators and Matrices We say that T is a Toeplitz matrix if T is a block matrix of the form ⎡ ⎤ F0 F−1 F−2 · · · ⎢ F1 F0 F−1 · · · ⎥ ⎢ ⎥ T = ⎢ F2 F1 . F0 · · · ⎥ ⎣ ⎦ .. .. . . 5. Toeplitz Operators and Matrices 35 Here {Fk }∞ −∞ is a sequence of operators mapping E into Y.

Using this, we obtain TΨ W = TΨ TΨ∗ TΘ = PM TΘ = TΘ . Hence TΨ W = TΘ . Using this along with the fact that both TΘ and TΨ intertwine the appropriate unilateral shifts, we obtain T Ψ SD W = SY T Ψ W = SY T Θ = T Θ SE = T Ψ W S E . Thus TΨ SD W = TΨ W SE . Because TΨ is one to one, W SE = SD W . 2. In other words, TΘ = TΨ W = TΨ TΩ . Therefore Θ(z) = Ψ(z)Ω for all z in D+ . 1. 2. As before, let M be an invariant subspace for the unilateral shift SY on 2+ (Y). Let Φ be any isometry mapping a space E into 2+ (Y) such that the range of Φ equals M SY M.

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