General Toeplitz matrices and the Takenaka-Malmquist basis Adhemar - PowerPoint PPT Presentation

General Toeplitz matrices and the Takenaka-Malmquist basis Adhemar Bultheel 1 and Pierre Carrette 2 August 2003 1 Department Computer Science 2 Shell Oil Company K.U.Leuven, Belgium Houston, Texas USA adhemar.bultheel@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ ∼ ade/ pierre.carrette@shell.com Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003

1/16 Takenaka-Malmquist basis Orthogonal rational functions w.r.t. Lebesgue measure on T � n − 1 1 − | α n | 2 z − α k � ϕ n ( z ) = , n ≥ 0 , 1 − α n z 1 − α k z k =0 � �� � B n ( z ) α k ∈ D , α 0 = 0 , ϕ − n = ϕ n (1 /z ) { ϕ n } k ∈ Z complete in L 2 ( T ) iff � (1 − | α k | ) = ∞ ⇒ B n ( z ) → 0 Assumption | α k | ≤ c < 1 , α 0 = 0 . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

2/16 Convergence in C q 2 π , q > 2 f ( ω ) := F ( e iω ) ∈ C q 2 π , q > 2 Classical theory = all α k = 0 : denote with a hat. � 2 π 1 ϕ k ( e iω ) = e ikω , ˆ ϕ k ( e iω ) dω ˆ c k = � f, ˆ ϕ k � = f ( ω ) ˆ 2 π 0 Thm: f ( ω ) = � ϕ k ( e iω ) absolutely convergent k ˆ c k ˆ 2 π , q > 2 then � Thm: f ∈ C q | k | <p c k ϕ k ( e iω ) → f ( ω ) , c k = � f, ϕ k � cvg uniform in ω ; rate of cvg is at least 1 /p q − 1 Proof is technical, based on | c 0 | ≤ � f � 1 and | c ± k | ≤ C � f ( q ) � 1 / ( ǫk ) q , k ≥ 1 C and ǫ depend on q , c , k Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

3/16 Example The function is f ( ω ) = (cos 2 ω + 2) / (4 cos 2 ω + 5) . √ f ( ω ) = Re F ( e iω ) , F ( z ) = ( z 2 + 2) − 1 . Poles at ± i 2 . 1 −3 x 10 14 0.9 12 0.8 10 0.7 8 0.6 6 0.5 4 0.4 2 0.3 0.2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure: left: f and f 2 , random zeros; right: f − f 4 and f − f 6 for poles α k = ( − 1) k 0 . 6 . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

4/16 Example convergence The function is f ( ω ) = (cos 2 ω + 2) / (4 cos 2 ω + 5) . √ f ( ω ) = Re F ( e iω ) , F ( z ) = ( z 2 + 2) − 1 . Poles at ± i 2 . 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 −8 −8 10 10 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Figure: left: � f − f p � ∞ for (1) α k random, (2) α k = 0 , (3) α k = ( − 1) k 0 . 6 , right: Fourier coefs for case (3) Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

5/16 Convergence in C 2 π Here convergence need not be uniform ⇒ Ces` aro means. Example: The function is f ( ω ) = | ω − π | . 0.35 0.25 0.3 0.2 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure: left: | f − f n | , n = 2 , 4 , 6 for poles α k = 0 . right: Ces` aro sum for n = 6 . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

6/16 Ces` aro sums and Toeplitz matrix � 1 � 2 π � 2 π � p − 1 = 1 ˆ Γ p ( ω ) f ( ω )ˆ ˆ e ikω f ( ω ) e ilω dω Γ ∗ M p ( f ) = p ( ω ) dω 2 π 2 π 0 0 k,l =0 Γ p ( ω ) = [1 , e − iω , . . . , e − i ( p − 1) ω ] ∗ = [ ˆ ˆ ϕ p − 1 ( e iω )] ∗ ϕ 0 ( e iω ) , . . . , ˆ Note p − 1 � ˆ p ( µ )ˆ ϕ k ( e iµ ) = ˆ Γ ∗ ϕ k ( e iσ ) ˆ Γ p ( σ ) = ˆ K p ( µ, σ ) k =0 γ p = ˆ p ( ω )ˆ Γ p ( ω ) = ˆ Γ ∗ ˆ K p ( ω, ω ) = p Recall � 2 π M p ( f ) = 1 ˆ Γ p ( ω ) f ( ω )ˆ ˆ Γ ∗ p ( ω ) dω 2 π 0 Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

7/16 � Set ˆ G p ( ω ) = ˆ Γ p ( ω ) / ˆ γ p . Then � � 1 − | k | � ˆ p ( ω ) ˆ M p ( f ) ˆ G ∗ ϕ k ( e iω ) G p ( ω ) = ˆ c k ˆ p | k | <p This is Ces` aro sum and ˆ p ( ω ) ˆ M p ( f ) ˆ G ∗ lim G p ( ω ) = f ( ω ) p →∞ ˆ p ( ω ) ˆ M p ( f ) ˆ M p ( g ) ˆ G ∗ lim G p ( ω ) = f ( ω ) g ( ω ) p →∞ Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

8/16 Generalized Ces` aro sums Assume f ∈ C q 2 π with q > 4 . � 2 π 1 Γ p ( ω ) f i ( ω )Γ ∗ Generalized Toeplitz: M p ( f i ) = p ( ω ) dω 2 π 0 Γ p ( ω ) = [ ϕ 0 ( e iω ) , . . . , ϕ p − 1 ( e iω )] ∗ G p ( ω ) = Γ p ( ω ) γ p ( ω ) = Γ ∗ , p ( ω )Γ p ( ω ) � γ p ( ω ) � T ( f 1 , . . . , f n ) , σ = µ G ∗ p ( σ ) T ( M p ( f 1 ) , . . . , M p ( f n )) G p ( µ ) = 0 , σ � = µ T ( x 1 , . . . , x n ) analytic function in n variables. Rate of convergence as fast as 1 /p if σ = µ or ln p p if σ � = µ . Depends on analysis of the generalized Toeplitz matrix M p ( f ) . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

9/16 Spectral properties Classical case � 2 π p − 1 1 M p ( f )) = 1 � λ i ( ˆ lim f ( ω ) dω p 2 π p →∞ 0 i =0 Generalized to f ∈ C q 2 π , q > 4 � 2 π p − 1 1 T ( λ i ( M p ( f ))) = 1 � � � f ( χ − 1 ( ω )) lim T dω p 2 π p →∞ 0 i =0 χ p ( ω ) χ ( ω ) = lim p →∞ , χ p ( ω ) = phase of B p ( ω ) . p Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

10/16 Example 0.605 0.6 0.595 f ( ω ) = cos 2 ω + 2 0.59 4 cos 2 ω + 5 0.585 0.58 0.575 0.57 2 4 6 8 10 12 14 16 18 20 � 2 π � 2 π p − 1 1 λ k ( M p ( f )) = 1 � χ − 1 ( ω )) dω = lim f (˜ f ( µ )˜ χ ( µ ) dµ p 2 π p →∞ 0 0 k =0 left-hand side for p = 20 is 0.6045, right-hand side integral = 0.6098 Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

11/16 Quadrature formulas The basic idea in the proofs is to approximate integrals by quadratures. � 2 π M p ( f ) = 1 Γ p ( ω ) f ( ω )Γ ∗ p ( ω ) dω 2 π 0 Use � 2 π p − 1 1 � g ( ω ) dω ≈ H k g ( ω k ) . 2 π 0 k =0 Then M p ( f ) ≈ W p ( f ) = Υ p F p Υ ∗ p F p = diag( f ( ω 0 ) , . . . , f ( ω p − 1 )) , � Υ p = [˜ Γ p ( ω 0 ) , . . . , ˜ ˜ Γ p ( ω p − 1 )] , Γ p ( ω k ) = Γ p ( ω k ) H k Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

12/16 The quadrature formula is exact in L p · L − p = span { ϕ k ( e iω ) ϕ l ( e iω ) : k, l = 0 , . . . , p − 1 } if ω k are zeros of para-orthogonals ϕ p ( e iω ) − ηB p ( e iω ) ϕ p ( e iω ) , η ∈ T and H k = 1 /γ p ( ω k ) . The classical case = Szeg˝ o quadrature: ω k equidistant, H k = 1 /p . Then Υ p is the (unitary) FFT matrix. M p ( f ) ≈ W p ( f ) = Υ p F p Υ ∗ p EVD of W p ( f ) ≈ M p ( f ) , so f ( ω k ) ≈ f p ( ω k ) approximate the eigenvalues of M p ( f ) . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

13/16 The general case = rational Szeg˝ o: ω k ∈ Ω p ( θ ) , H k = 1 /γ p ( ω k ) . Ω p ( θ ) = { ω ∈ [0 , 2 π ) : χ p ( ω ) = θ mod 2 π ; η = e iθ ; B p ( e iω ) = e iχ p ( ω ) } . 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figures: χ 10 ( ω ) , left α k = ± 0 . 6 i , right α k = 0 . 9 e ikπ/ 10 Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

14/16 Spectral approximation 0.3 1 0.9 0.25 0.8 0.2 0.7 0.15 0.6 0.5 0.1 0.4 0.05 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 f ( ω ) = (cos 2 ω +2) / (4 cos 2 ω +5) . Left: 2-norm of the vector of differences λ i ( M p ( f )) − f ( ω i ) as a function of p . Right diag(Υ ∗ p M p ( f )Υ p ) (circles) and f ( ω ) . Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

15/16 Spectral approximation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cumulative distribution of f ( ω k ) for ω k ∈ Ω 20 (1) (dashed) and cumulative distribution of λ k ( M 20 ( f )) (solid). Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel

16/16 Reference [1] A. Bultheel, P. Carrette. Algebraic and spectral properties of general Toeplitz matrices. SIAM J. Control Optim. , 41 (2003), pp. 1413–1439. Orthogonal Functions and Related Topics, Røros, Norway, Augist 12-16, 2003 A. Bultheel