Analytic Interpolation on the Unit Disk History, Recent Developments, and Applications Hendra Ishwara Nurdin Department of Information Engineering, Research School of Information Sciences and Engineering (RSISE), Australian National University, Canberra ACT 0200, Australia Analytic Interpolation on the Unit Disk – p. 1/39
Contents • Orthogonal polynomials • The Carathèodory and Nevanlinna-Pick interpolation problem • The Schur algorithm • Rational interpolation with a degree bound • The Kimura-Georgiou parametrization Analytic Interpolation on the Unit Disk – p. 2/39
Contents (cont’d) • Parametrization of all bounded degree solutions • Computing bounded degree solutions • Some engineering applications: Circuit theory, spectral estimation, and spectral factorization Analytic Interpolation on the Unit Disk – p. 3/39
Orthogonal Polynomials • A simple but powerful concept introduced by Szëgo. • Let C = { c 0 , c 1 , c 2 , ... } be a bonafide covari- ance sequence of a stochastic process, i.e. c 0 c 1 c 2 . . . c n . ... . . c 1 c 0 c 1 ... c 2 T n = ≥ 0 for all n c 2 c 1 c 0 . ... ... ... c 1 . . c n . . . c 2 c 1 c 0 Analytic Interpolation on the Unit Disk – p. 4/39
Orthogonal Polynomials (cont’d) • { c 0 , c 1 , . . . } induces an inner product on the space of polynomials with complex coefficients: u v u v b j z j > = � � � � a i z i , < a i c j − i b j i =0 j =0 i =0 j =0 where c − t = c t . • Is a norm (semi-norm) if sequence is positive (non-negative). Analytic Interpolation on the Unit Disk – p. 5/39
Orthogonal Polynomials (cont’d) • By Gram-Schmidt, one can construct an orthogonal set of polynomials { Φ 0 = 1 , Φ 1 , Φ 2 , . . . } from the standard basis { 1 , z, z 2 , . . . } , called orthogonal polynomials of the first kind . • The Φ i ’s satisfy a certain recurrence relation: Φ k ( z ) = z Φ k − 1 − r k Φ k − 1 ( z ) # k = 1 , 2 , ... where Φ i ( z ) # , the reversed polynomial , is defined by Φ i ( z ) # = z deg(Φ i ) Φ i (¯ z − 1 ) . Analytic Interpolation on the Unit Disk – p. 6/39
Orthogonal Polynomials (cont’d) • { r 1 , r 2 , . . . } are called reflection or Schur coefficients, they satisfy | r i | ≤ 1 . • If C is positive then | r i | < 1 ∀ i ≥ 1 , while if C is non-negative but not positive then | r i | < 1 for i = 1 , . . . , s − 1 and | r s | = 1 with s = min { k || T k | = 0 } . • Bijectivity exists between positive C ’s and pairs ( c 0 , R ) with c 0 > 0 and R a sequence with terms having modulus less than 1. Analytic Interpolation on the Unit Disk – p. 7/39
Orthogonal Polynomials (cont’d) • Given C then ( c 0 , R ) can be computed and vice versa. It is also true that if C is positive then the first n + 1 terms of C determine c 0 and the first n terms of R and vice versa, • There is also a bijectivity between non-negative sequences C with s = min { k || T k | = 0 } < ∞ and pairs ( c 0 , R s ) with c 0 > 0 and R s = { r 1 , . . . , r s } satisfying | r k | < 1 for k = 1 , · · · , s − 1 and | r s | = 1 . In this case, and in this case alone, c 0 , c 1 , . . . , c n has a unique extension which is rational. Analytic Interpolation on the Unit Disk – p. 8/39
Orthogonal Polynomials (cont’d) 1 • If | r i | < 1 ∀ i ≥ 1 then Φ n ( z ) converges as n → ∞ to a spectral factor of the spectral density associated with c 0 , c 1 , . . . . Analytic Interpolation on the Unit Disk – p. 9/39
Notation • Let H ( D ) denote the class of all functions holomorphic in D . • Let C = { f ∈ H ( D ) : ℜ f ( z ) ≥ 0 ∀ z ∈ D } and let C A + denote the subset of C containing functions which are positive ( > 0) on ∂ D and with Taylor coefficients in A ⊂ C . • C is typically known as the Carathèodory class . Analytic Interpolation on the Unit Disk – p. 10/39
The General Interpolation Problem • Let there be given an indexed set Z = { z 1 , . . . , z n } ⊂ D and W = { w 1 , . . . , w n } ⊂ C . Assume the indexing in { z 1 , . . . , z n } is such that all non-distinct points are ordered consecutively. • Problem: Given ( Z , W ) find all f ∈ C such that: f ( z k ) = w k if # z k = 1 , or f ( l ) ( z k + l ) = w k + l l = 0 , . . . , m − 1 if # z k = m Analytic Interpolation on the Unit Disk – p. 11/39
The Carathèodory Extension Problem • Special case of the general problem if z 0 = . . . = z n = 0 , w 0 = 1 2 c 0 , and w i = c i for i = 1 , . . . , n . • It is also known as the covariance extension problem . • Has a solution if and only if the Toeplitz matrix T n = [ c j − i ] n i,j =1 ≥ 0 where c − i = c i . • Extension is unique and is a linear combina- tion of sinusoids iff T n is non-negative but singular. Analytic Interpolation on the Unit Disk – p. 12/39
Nevanlinna-Pick Interpolation • Special case of the general problem if z 0 , . . . , z n are all distinct (i.e. each has multiplicity 1). • Has a solution if and only if the Pick matrix P = [ w i + w j 1 − z i z j ] n i,j =1 is nonnegative definite. • Solution is unique and is a linear combination of sinusoids iff P is non-negative but singular. Analytic Interpolation on the Unit Disk – p. 13/39
The Schur Class S • S is the class of all functions S holomorphic in D satisfying | S ( z ) | ≤ 1 ∀ z ∈ D . • There is an LFT which relates an element of C to an element of S given by: S ( z ) = 1 C ( z ) − C (0) 2 C ( z ) + C (0) where S ∈ S while C ∈ C . • In fact, there is bijectivity between C and S . Analytic Interpolation on the Unit Disk – p. 14/39
The Schur Algorithm • A famous algorithm attributed to Schur. • Idea: A function S 0 is in S iff S 0 ( z ) − S 0 (0) 1. | S 0 (0) | <1 and S 1 ( z ) = 1 1 − S 0 (0) S 0 ( z ) is in S , or z 2. S 0 is a constant function of modulus 1. then repeat the iteration for S 1 , S 2 , . . . until S n is a constant function of modulus one, or keep going forever. • S 1 (0) , S 2 (0) , . . . are called the Schur coeffi- cients . Analytic Interpolation on the Unit Disk – p. 15/39
Solving the Carathèodory Extension Problem • Translating the result of Schur to C : A function C 0 beginning with 1 + 2 c (0) 1 z is in C iff 1. | c (0) s 0 ( z ) C 0 ( z ) − q 0 ( z ) 1 | < 1 and C 1 ( z ) = − r 0 ( z ) C 0 ( z )+ p 0 ( z ) is in C , with p 0 ( z ) = (1 + z )(1 − c (0) 1 ) , q 0 ( z ) = (1 − z )(1 − c (0) 1 ) , r 0 ( z ) = (1 − z )(1 + c (0) 1 ) , and s 0 ( z ) = (1 + z )(1 + c (0) 1 ) , or 1 | = 1 and C 0 ( z ) = 1+ c (0) 2. | c (0) 1 z 1 z . 1 − c (0) Analytic Interpolation on the Unit Disk – p. 16/39
Solving the Carathèodory Extension Problem(cont’d) • Compute c ( i ) 1 from i = 1 until i = n or until some u ≤ n such that c ( u ) = 1 . Then proceed 1 by: 1. If | c ( i ) 1 | < 1 for i = 1 , . . . , n , do the backward iteration: C i ( z ) = p i ( z ) C i +1 ( z )+ q i ( z ) r i ( z ) C i +1 ( z )+ s i ( z ) for i = n, . . . , 1 , with C n +1 is any function in C , otherwise 2. Set C u ( z ) = 1+ c ( u ) 1 z 1 z , proceed as in step (1) 1 − c ( u ) for i = u − 1 , . . . , 1 . Analytic Interpolation on the Unit Disk – p. 17/39
Solving the Carathèodory Extension Problem(cont’d) • The sequence r 1 , . . . , r n obtained via the orthogonal polynomials is exactly the same as the sequence c (1) 1 , . . . , c ( n ) 1 , hence also the name Schur coefficients for r 1 , . . . , r n ! Analytic Interpolation on the Unit Disk – p. 18/39
Solving the Nevanlinna-Pick Problem • It is solved by an algorithm akin to (or a variant of) Schur’s which was independently developed by Pick and Nevanlinna. • This algorithm is often referred to as the Nevanlinna-Schur (NS) algorithm. • NS recursion in S : If f k ∈ S then f k +1 ( z ) = 1 − z k z f k ( z ) − f k ( z k ) ∈ S z − z k 1 − f k ( z k ) f k ( z ) Analytic Interpolation on the Unit Disk – p. 19/39
Introducing a Degree Bound • Dewilde & Dym in 1981: In a particular situation one can always construct a rational function in C of (McMillan) degree ≤ n satisfying the general interpolation constraints. • Q: Is this existence more general? If so, can one parametrize all such solutions in a system theoretically “meaningful” way? • Q: How can one explicitly compute all such rational solutions? Analytic Interpolation on the Unit Disk – p. 20/39
Introducing a Degree Bound (cont’d) • Thus investigation into rational solutions of the covariance extension problem with degree ≤ n started. • Name of new problem: Rational covariance extension problem (RCEP). • The constraint on the degree of the solution also known as "complexity constraint". Degree = complexity. Analytic Interpolation on the Unit Disk – p. 21/39
The Kimura-Georgiou Parametrization • Kimura & Georgiou: All rational covariance interpolators in H ( D ) (but not necessarily in C ) of degree at most n has the form (with c 0 normalized to 1): f ( z ) = 1 Ψ n + α 1 Ψ n − 1 + ... + α n Ψ 0 2 Φ n + α 1 Φ n − 1 + ... + α n Φ 0 with α i ∈ C and { Ψ i } i ≥ 1 another set of orthogonal polynomials (of the second kind). • Hence all rational covariance extensions are proper, but never strictly proper! Analytic Interpolation on the Unit Disk – p. 22/39
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