A survey of optimal polynomial approximants, applications to digital filter design, and related open problems
AA SURVEY OF OPTIMAL POLYNOMIALAPPROXIMANTS, APPLICATIONS TO DIGITALFILTER DESIGN, AND RELATED OPEN PROBLEMS
CATHERINE B´EN´ETEAU AND RAYMOND CENTNER
Abstract.
In the last few years, the notion of optimal polyno-mial approximant has appeared in the mathematics literature inconnection with Hilbert spaces of analytic functions of one or morevariables. In the 70s, researchers in engineering and applied math-ematics introduced least-squares inverses in the context of digitalfilters in signal processing. It turns out that in the Hardy space H these objects are identical. This paper is a survey of knownresults about optimal polynomial approximants. In particular, wewill examine their connections with orthogonal polynomials andreproducing kernels in weighted spaces and digital filter design.We will also describe what is known about the zeros of optimalpolynomial approximants, their rates of decay, and convergenceresults. Throughout the paper, we state many open questions thatmay be of interest. Introduction
Consider a collection of Hilbert spaces of analytic functions of theunit disk D defined as follows. Given weights { ω k } k ∈ N satisfying ω = 1, ω k > ω k /ω k +1 → k → ∞ , we define the weighted Hardyspace H ω as the space of functions f ( z ) = (cid:80) ∞ k =0 a k z k analytic in D such that (cid:107) f (cid:107) ω = ∞ (cid:88) k =0 | a k | ω k < ∞ .H ω is a Hilbert space with inner product (cid:104) f, g (cid:105) ω = ∞ (cid:88) k =0 a k b k ω k , for g ( z ) = (cid:80) ∞ k =0 b k z k . If ω k = ( k +1) α for a real parameter α, the spaces H ω = D α are called Dirichlet-type spaces and include the Hardy space Mathematics Subject Classification.
Primary 30Hxx; Secondary 30Jxx.Key words: optimal polynomial approximants, digital filters, Dirichlet spaces, re-producing kernels, orthogonal polynomials. a r X i v : . [ m a t h . C V ] F e b B´EN´ETEAU AND CENTNER H = D , the classical Dirichlet space D = D , and the Bergman space D − = A . In the case of the Hardy space H , we omit the subscript ω and write (cid:104)· , ·(cid:105) for the inner product. Given a function f ∈ H ω not identically 0 and n ∈ N , we define the n -th optimal polynomialapproximant (opa) of 1 /f in H ω to be the polynomial q n that minimizes (cid:107) pf − (cid:107) ω among all polynomials p of degree at most n. Notice that ifwe denote by P n to be the set of all polynomials of degree at most n ,then f · P n is a finite dimensional subspace of H ω , and q n f is simplythe projection of 1 onto that subspace. Therefore, q n always exists andis unique.These polynomials were defined in [1] in the spaces D α in connectionwith cyclicity: a function f ∈ H ω is called cyclic if its polynomialmultiples are dense in the space. In particular, if f is cyclic, theremust be a sequence of polynomial multiples of f that approximate 1,and thus, the opas q n are the ones that achieve this approximationat the fastest rate in terms of the degree. On the other hand, if 1can be approximated by polynomial multiples of f , then so can anypolynomial, and since polynomials are dense in H ω , f must be cyclic.Therefore f is cyclic if and only if the opas q n satisfy (cid:107) q n f − (cid:107) ω → n → ∞ . Thus, one can view cyclicity through the lens of the behaviorof the optimal polynomial approximants.The authors of [1] were hoping to be able to shed some light on theBrown and Shields conjecture (see [9]) using opas. The Brown andShields conjecture states that for a function f in the classical Dirichletspace D , if f is outer and has boundary zero set Z ( f ) with logarith-mic capacity 0, then f is cyclic. (The converse is known to be true.)Although this initial hope has not yet proved successful, many other in-teresting problems related to optimal polynomial approximants arose,in particular in connection with inner functions (see, e.g., [3, 15, 24]).It turns out that optimal polynomial approximants had been intro-duced much earlier in the engineering literature in the 70s, in connec-tion with signal processing and digital filter construction. They werefirst introduced by Robinson ([23]) and referred to as least-squares in-verses. Robinson wanted to find the inverse under discrete convolutionof a finite-length minimum-delay wavelet that best approximates theunit spike. From a mathematical point of view, this meant, given afinite sequence b := ( b , b , . . . , b n ) of real numbers, where the largestcoefficient in magnitude is b (the so-called “minimum delay”), find a“wavelet”, i.e., a sequence a := ( a , a , . . . , a m ) of real numbers suchthat the difference between the convolution a ∗ b and the unit spike(1 , , . . . ,
0) has smallest (cid:96) -norm. This is equivalent to the problem offinding the optimal polynomial approximant of 1 /f for a polynomial f URVEY OF OPTIMAL APPROXIMANTS 3 with f (0) (cid:54) = 0. In [12], Chui examined the double least-squares inverse of a polynomial f of degree n by first finding the k -th opa q k of 1 /f and then finding the n -th opa p n,k of 1 /q k . He studied the zeros of thedouble least-squares inverses, noticed the connection with orthogonalpolynomials (which had also been discussed for polynomials with realcoefficients in [17]), and proved that p n,k converges to f in the finite-dimensional space of polynomials of degree n if and only if f has nozeros in the open unit disk. He went on to discuss some generalizationsto the case when f ∈ H , which was further studied by Izumino in [20].In several variables, least-squares inverses were studied by a number ofauthors in the 70s in connection with recursive digital filters (see, e.g.,[17, 27]). Two recent papers of Sargent and Sola ([25, 26]) discuss someof these developments and examine the more complicated relationshipbetween optimal polynomial approximants and orthogonal polynomialsin several variables.This paper is a survey of some of the known results about optimalpolynomial approximants, applications to digital filter theory, and re-lated open problems. One of the goals of the paper is, as also discussedin [25], to connect the two sets of literature on these topics. In Section2, we describe how to compute opas, give some basic examples, andexplain how opas are connected to orthogonal polynomials and repro-ducing kernels for weighted spaces. In Section 3, we discuss how opaswere used in the design of digital filters in the engineering literature ofthe 70s. In Section 4, we survey some known results about the zeros ofopas in the single and multivariable case and state several open ques-tions. Finally, in Section 5, we discuss rates of decay and pointwiseconvergence results.2. Existence of opas and connections with orthogonalpolynomials and reproducing kernels
Let us now turn to a discussion of how to compute opas and theirconnection with classical objects in analysis. We have already seenthat for any function f ∈ H ω \ { } , and any n ∈ N , the n -th opa q n always exists. It is also not hard to see that the coefficients of q n canbe computed by a set of linear equations ([1, 16]), as stated in thefollowing theorem. Theorem 2.1.
For any function f ∈ H ω \ { } , let q n ( z ) = (cid:80) nj =0 a j z j denote the n -th opa of /f . Define the vectors a = (cid:2) a a . . . a n (cid:3) T and y = (cid:2) f (0) 0 . . . (cid:3) T . If B is the ( n + 1) × ( n + 1) matrix given B´EN´ETEAU AND CENTNER by B jk = (cid:104) z k f, z j f (cid:105) ω , where ≤ j, k ≤ n , then Ba = y. Proof.
Note that by definition, q n is the projection of 1 onto P n · f . Byorthogonality in H ω , we thus have that0 = (cid:104) q n f − , z j f (cid:105) ω = (cid:104) n (cid:88) k =0 a k z k f − , z j f (cid:105) ω = n (cid:88) k =0 a k (cid:104) z k f, z j f (cid:105) ω − (cid:104) , z j f (cid:105) ω = n (cid:88) k =0 a k (cid:104) z k f, z j f (cid:105) ω − δ j, f (0)for j = 0 , · · · , n . This is equivalent to the assertion Ba = y . (cid:3) Let’s have a look at an example for H in which the form of thematrix is simple enough to allow us to determine an explicit formulafor the coefficients. Example 2.2.
Let q n denote the n -th opa of 1 /f , where f ( z ) = 1 − z .Then q n ( z ) = n (cid:88) j =0 n + 1 − jn + 2 z j . Proof.
Let q n ( z ) = (cid:80) nj =0 a j z j and let a and y denote the vectors a = (cid:2) a a a . . . a n (cid:3) T and y = (cid:2) . . . (cid:3) T . From Theorem2.1, we have Ba = y, where B is the ( n + 1) × ( n + 1) matrix with B j,k = (cid:104) z k f, z j f (cid:105) = (cid:104) z k (1 − z ) , z j (1 − z ) (cid:105) = (cid:104) z k − z k +1 , z j − z j +1 (cid:105) = (cid:104) z k , z j (cid:105) + (cid:104) z k +1 , z j +1 (cid:105) − (cid:104) z k +1 , z j (cid:105) − (cid:104) z k , z j +1 (cid:105) for 0 ≤ j, k ≤ n . Therefore, we see that B j,k = j = k − | j − k | = 10 otherwise URVEY OF OPTIMAL APPROXIMANTS 5 and note that a = 1 / n = 0. If n = 1, we get (cid:40) a − a = 1 − a + 2 a = 0 . For n ≥
2, the coefficients { a j } nj =0 satisfy a − a = 1 − a j + 2 a j +1 − a j +2 = 0 0 ≤ j ≤ n − − a n − + 2 a n = 0 . These conditions can be summarized by the matrix equation − . . . − − . . . − − . . . . . . − − . . . − a a a a ... a n = . It is straightforward to check that { n +1 − jn +2 } nj =0 satisfies these conditions. (cid:3) This computation was generalized in [16], where the authors showthat the opa q n of 1 / (1 − z ) in H ω is equal to q n ( z ) = n (cid:88) k =0 (cid:32) − (cid:80) kj =0 1 ω j (cid:80) n +1 j =0 1 ω j (cid:33) . In H , formulas for the opa of 1 / (1 − z ) a where Re ( a ) > f is a polynomialare considered in [7]. From the point of view of classical analysis, itis easy to see that one way to calculate q n f is to consider the set ofvectors { f, zf, z f, . . . , z n f } , use the Gram-Schmidt process to find anorthonormal basis of the finite dimensional space f · P n , and express q n f in that basis. This gives rise to orthonormal polynomials and isthe content of the following theorem, discussed in [5]. Theorem 2.3 ([5]) . Let f ∈ H ω \ { } , n ∈ N , and let q n be the n -thopa of /f. For each k = 0 , , . . . n, let ϕ k be a polynomial of degree k such that { ϕ k f } nk =0 is an orthonormal basis of f · P n . Then q n ( z ) = f (0) n (cid:88) k =0 ϕ k (0) ϕ k ( z ) . (1) B´EN´ETEAU AND CENTNER
Proof.
Since q n f ∈ f · P n and { ϕ k f } nk =0 is an orthonormal basis of f · P n , we can express q n f in that basis: q n ( z ) f ( z ) = n (cid:88) k =0 (cid:104) q n f, ϕ k f (cid:105) ω ϕ k ( z ) f ( z ) . But q n f is the projection of 1 onto f · P n , and therefore (cid:104) q n f, ϕ k f (cid:105) ω = (cid:104) , ϕ k f (cid:105) ω . By the defintion of the inner product in H ω , this gives q n ( z ) f ( z ) = n (cid:88) k =0 f (0) ϕ k (0) ϕ k ( z ) f ( z ) , and therefore, (1) follows. (cid:3) The polynomials ϕ k can be thought of as orthonormal polynomialsin the Hilbert space weighted by f , in the sense that (cid:104) ϕ k f, ϕ j f (cid:105) ω = δ kj . In particular, if we restrict to the classical space H , this is the sameas saying that the polynomials ϕ k are orthonormal polynomials in theweighted space H ( µ ) where the measure dµ = π | f | dθ. This approachwas discussed in [13] for the Hardy space and in [5] for Dirichlet-typespaces.Notice also that since { ϕ k f } nk =0 is an orthonormal basis of f · P n , (cid:80) nk =0 f (0) ϕ k (0) ϕ k ( z ) f ( z ) =: K n ( z,
0) is the reproducing kernel for f ·P n at 0 : this means that for any polynomial q ∈ P n , (cid:104) qf, K n ( · , (cid:105) ω = q (0) f (0) . This can also be seen directly, since (cid:104) qf, q n f (cid:105) ω = (cid:104) qf, (cid:105) ω = q (0) f (0) . Thus q n ( z ) f ( z ) = K n ( z, . In the particular case of H , the n -th opa q n is even more tightlyconnected to the n -th orthonormal polynomial, as is well-known (see,e.g., [18, 28]). The authors of [5] used this connection to express theopas in terms of weighted orthonormal polynomials. Theorem 2.4 ([5]) . Let f ∈ H \ { } , n ∈ N , and let q n be the n -th opaof /f. For each k = 0 , , . . . n, let ϕ k be a polynomial of degree k suchthat { ϕ k f } nk =0 is an orthonormal basis of f ·P n . Let ϕ ∗ n ( z ) = z n ϕ n (1 / ¯ z ) . Then q n ( z ) = f (0) ˆ ϕ n ( n ) ϕ ∗ n ( z ) . URVEY OF OPTIMAL APPROXIMANTS 7
Proof.
Let q n ( z ) = (cid:80) nk =0 a k z k and consider the weighted space H ( µ ),where dµ = π | f | dθ . In light of Theorem 2 .
1, we see that δ j, f (0) = n (cid:88) k =0 (cid:104) z j f, z k f (cid:105) a k = (cid:104) z j f, n (cid:88) k =0 a k z k f (cid:105) = (cid:104) z j , q n (cid:105) µ = (cid:104) z n q n , z n − j (cid:105) µ = (cid:104) q ∗ n , z n − j (cid:105) µ . Now since { ϕ k } nk =0 is an orthonormal basis of H ( µ ), it follows that q ∗ n ( z ) = n (cid:88) k =0 (cid:104) q ∗ n , ϕ k (cid:105) µ ϕ k ( z )= (cid:104) q ∗ n , ϕ n (cid:105) µ ϕ n ( z )= f (0) ˆ ϕ n ( n ) ϕ n ( z ) . Therefore, q n ( z ) = z n q ∗ n (1 /z )= f (0) ˆ ϕ n ( n ) z n ϕ n (1 /z )= f (0) ˆ ϕ n ( n ) ϕ ∗ n ( z ) . (cid:3) It is known from the theory of orthogonal polynomials that the or-thonormal polynomials ϕ n have all their zeros inside the open unit disk D , and therefore the opas q n have no zeros in the closed disk. In par-ticular, this connection was important for designing what are calledstable filters, which we discuss in the next section.3. Digital Filter Design
Introduction to filters.
Several problems in engineering ulti-mately depend on a system’s response to an input. In the case of adigital system, the input is given as a sampling sequence { x ( n ) } ∞ n = −∞ ,and the output { y ( n ) } ∞ n = −∞ can often be described by a differenceequation y ( n ) = M (cid:88) k =0 b k x ( n − k ) − N (cid:88) j =1 a j y ( n − j ) , (2) B´EN´ETEAU AND CENTNER where the coefficients a j and b k are real numbers.If the coefficients remain constant over time, the system is knownas linear time-invariant , or LTI. If the a j ’s are not all zero, then thesystem is referred to as recursive . This means that one or more ofthe system’s output is used as an input. Now, if we consider an inputsequence { x ( n ) } ∞ n = −∞ that is bounded, it seems problematic in practicefor | y ( n ) | to increase without bound as n → ∞ . Therefore, it is ofinterest to seek for properties of a system that preserve boundedness.A system in which a bounded input yields a bounded output is called BIBO stable .In order to facilitate our discussion of filters, we will assume thatour input sequences { x ( n ) } ∞ n = −∞ have the property that x ( n ) = 0 for n <
0. A sequence with this property is known as causal . Moreover, wewill assume that the input sequences are exponentially bounded . Thatis, we assume that | x ( n ) | ≤ K n , n ≥ n for some constant K and some integer n . Now, to better understandthe relationship between the input and output sequences, we make useof the following operator. For any causal sequence { a ( n ) } ∞ n = −∞ that’sexponentially bounded, consider the mapping { a ( n ) } ∞ n = −∞ (cid:55)→ ∞ (cid:88) n =0 a ( n ) z − n . This mapping is known as the z-transform . It is a linear operator fromthe space of exponentially bounded causal sequences onto the space offunctions analytic at ∞ . The z -transform of a sequence { a ( n ) } ∞ n = −∞ has a region of convergence (ROC) given by z ∈ ˆ C such thatlim sup n (cid:112) | a ( n ) | < | z | . (3)With this, the sum and product of two transformed sequences are de-fined to be in the intersection of both ROCs, and the product is givenby the expression (cid:32) ∞ (cid:88) n =0 a ( n ) z − n (cid:33)(cid:32) ∞ (cid:88) n =0 b ( n ) z − n (cid:33) = ∞ (cid:88) n =0 (cid:32) n (cid:88) k =0 a ( k ) b ( n − k ) (cid:33) z − n . The z -transform has many properties that make it useful in theanalysis of digital systems. In particular, if A ( z ) is the z -transformof the sequence { a ( n ) } ∞ n = −∞ , then the z -transform of the sequence { c ( n ) = a ( n − N ) : −∞ < n < ∞} is z − N A ( z ) for any N ∈ N . There-fore, by applying the z -transform to both sides of (2), we see that the URVEY OF OPTIMAL APPROXIMANTS 9 z -transforms of the input and output are related by the equation Y ( z ) = H ( z ) X ( z ) , where H ( z ) is given by the rational function H ( z ) = (cid:80) Mk =0 b k z − k (cid:80) Nj =1 a j z − j . (4)We use X ( z ) and Y ( z ) to represent the the z -transforms of the inputand output sequences, respectively. The rational function H is knownas the transfer function, or filter , of the system. In the case of a re-cursive system, the transfer function is commonly called an infiniteimpulse response filter , or IIR filter.For the purpose of our discussion, we will only be considering recur-sive LTI systems. For simplicity, we refer to a BIBO stable system as stable . Likewise, we refer to the filter of a BIBO stable system as stable.The goal of Section 3 is to demonstrate how optimal polynomial ap-proximants are used in designing a stable filter. It’s worth noting thatalthough we will be considering filters of a single variable, several appli-cations are concerned with filters of multiple variables. As mentionedin [19], the processing of medical pictures, satellite photographs, radarand sonar maps, seismic data mappings, gravity waves data, and mag-netic recordings are examples in which 2D signal processing is needed.Here, we are concerned with designing filters H ( z, w ) in C of analogousstability.3.2. Stability.
An important part of designing a digital filter is ensur-ing that the filter is stable. The stability of the filter will prevent themagnitude of the output from increasing without bound, which couldbe damaging to the physical system. Therefore, we seek for proper-ties of the expression in (4) which guarantee stability. If we define thepolynomials A ( z ) = 1 + (cid:80) Nj =1 a j z j and B ( z ) = (cid:80) Mk =0 b k z k , we see that H ( z ) = z N − M B ∗ ( z ) A ∗ ( z ) , (5)where A ∗ ( z ) and B ∗ ( z ) are the reverse polynomials of A ( z ) and B ( z ),respectively. From (5), it is easy to check that H ( z ) is a rationalfunction analytic at ∞ . Therefore, H ( z ) is the z -transform of somecausal sequence { h ( n ) } ∞ n = −∞ , and we write H ( z ) = ∞ (cid:88) n =0 h ( n ) z − n (6)for some specified ROC. Now if we assume that the poles of H ( z ) are contained in the disk D , the expression in (6) must be vaild on T . Since the series definedby (cid:80) ∞ n =0 | h ( n ) | z − n has the same ROC, it follows that ∞ (cid:88) n =0 | h ( n ) | < ∞ . (7)This observation leads us to the following theorem. Theorem 3.1.
A filter H ( z ) is stable if its poles are contained in thedisk D .Proof. Let { x ( n ) } ∞ n = −∞ be an input sequence with | x ( n ) | ≤ M for all n .If the poles of H ( z ) are contained in D , then (7) holds. Consequently,the output sequence { y ( n ) } ∞ n = −∞ is bounded with | y ( n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) k =0 h ( k ) x ( n − k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M ∞ (cid:88) n =0 | h ( n ) | for all n . By definition, H ( z ) must be stable. (cid:3) This theorem gives a sufficient condition for a filter to be stable.However, it’s important to note that a filter with a pole outside of D need not be stable. As an example, consider the function H ( z ) = 1 z − . For all | z | >
1, this function is represented by the series H ( z ) = ∞ (cid:88) n =1 z − n . If we consider the input defined by x ( n ) = 1 for n ≥
0, the magnitudeof the output is given by | y ( n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) k =0 h ( k ) x ( n − k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n, which clearly increases without bound as n → ∞ . URVEY OF OPTIMAL APPROXIMANTS 11
Frequency Response.
Many problems are concerned with howa system responds to a sinusoidal input. This is particularly evident inaudio equalizing, where the input function represents a superpositionof multiple sound waves. Under the assumption that the system islinear, it is therefore advantageous to study the response of a systemto the input x ( n ) = e ins , where s ∈ R is a particular frequency. In thiscase, the output is known as the frequency response of the system. Ifwe let H ( z ) be the corresponding filter, then the frequency response isexpressed as y ( n ) = n (cid:88) k =0 h ( k ) e i ( n − k ) s = (cid:20) n (cid:88) k =0 h ( k ) e − iks (cid:21) e ins = H ( e is ) e ins . We then see that the frequency response is bounded, with | y ( n ) | = | H ( e is ) | , n ≥ . The quantity | H ( e is ) | is referred to as the magnitude of the frequencyresponse . To get an idea of what this function looks like, consider thefilter H ( z ) = 0 . z + 2 z + 1)1 . z + 1 . The poles and zeros of H ( z ) are displayed in the following diagram: The poles of H ( z ) are marked with a cross and the zero of H ( z ) ismarked with a circle. On the interval [0 , π ], we therefore expect | H ( e is ) | to have a maximum around 1.5 radians and a minimum around 3.1radians. This can be seen in the following graph:Often in the design of a digital filter, the goal is to develop a rationalfunction H ( z ) in which the modulus satisfies a set of specifications onthe boundary T . The effect of this would control the response of thesystem to the input x ( n ) = e ins . For the purpose of our discussion,we will assume that the specifications are given in the form of a non-negative even step function on [ − π, π ]. Such a step function is knownas an ideal digital filter . The problem of digital filter design can thenbe stated as follows: Problem 3.2.
For a given ideal filter χ ( e is ) , find a rational function H ( z ) with poles inside of D such that | H ( e is ) | is an approximation of χ ( e is ) . Methods of approximation which guarantee stability of the filter isan interesting topic of research. We present a method which has beenmodified from the ideas in [13]. This method creates a rational func-tion p ( z ) /q ( z ), with poles in D , such that | p ( e is ) | / | q ( e is ) | approximates χ ( e is ) in the least-squares sense . For any f ∈ L and any η >
0, wesay that the quotient g/h of two functions in L approximates f in theleast-squares sense if (cid:107) hf − g (cid:107) L < η. In this case, we call g/h an (LS)-approximant of f and write f ≈ LS g/h . URVEY OF OPTIMAL APPROXIMANTS 13
We will present the method of approximation in three stages. In thefirst stage, we will approximate the ideal filter by the magnitude of anon-vanishing function in H . In the second stage, we will use optimalpolynomial approximants to approximate this non-vanishing functionwith the magnitude of a rational function. In the third stage, we willalter the numerator and denominator of the rational function in orderto ensure stability.3.4. First stage of approximation.
We start the first stage by defin-ing a continuous function χ ε ( e is ) in the following way. Let S = { s j } Nj =1 denote the points of discontinuity of χ ( e is ). For each s j ∈ S , let I j = ( s j − ε/ , s j + ε/ ε is a positive number chosen so that theintervals do not overlap and such that ε is smaller than the minimumof the step values. If s / ∈ ∪ Nj =1 I j , set χ ε ( e is ) := (cid:40) χ ( e is ) if χ ( e is ) > ε if χ ( e is ) = 0 . This creates a positive step function on [ − π, π ] \ ∪ Nj =1 I j . Then con-nect each successive step with a straight line segment. For each s ∈∪ Nj =1 I j , set χ ε ( e is ) to coincide with these segments. This creates anon-vanishing continuous function on [ − π, π ].We then create an analytic function on D by using χ ε ( e is ). For any z ∈ D , define the function f ε ( z ) := exp (cid:18) π (cid:90) π − π e is + ze is − z log χ ε ( e is ) ds (cid:19) . Note that f ε ( z ) is analytic in D , non-vanishing in D , and has the prop-erty that log | f ε ( z ) | = 12 π (cid:90) π − π Re (cid:18) e is + ze is − z (cid:19) log χ ε ( e is ) ds. i.e., log | f ε ( z ) | solves the Dirichlet problem in D with boundary valuesdefined by log χ ε ( e is ). Therefore, the analytic function satisfies | f ε ( e is ) | = χ ε ( e is ) (8)for all s ∈ [ − π, π ]. This leads us to the following theorem. Theorem 3.3.
Any ideal filter χ can be approximated (in the least-squares sense) by a non-vanishing function f ∈ H . Proof.
Let χ ( e is ) be an ideal filter with a collection of discontinuities S = { s j } Nj =1 . Given any η >
0, choose ε to satisfy0 < ε < min (cid:26) η πN ( (cid:107) χ (cid:107) ∞ + 1) , η √ (cid:27) . Let { E k } k denote the collection of intervals for which χ ε ( e is ) = ε . From(8), it follows that (cid:107) χ − | f ε |(cid:107) L = (cid:107) χ − χ ε (cid:107) L = N (cid:88) j =1 π (cid:90) I j | χ ( e is ) − χ ε ( e is ) | ds + (cid:88) k π (cid:90) E k | χ ( e is ) − χ ε ( e is ) | ds ≤ ε N π (cid:107) χ (cid:107) ∞ + ε (cid:88) k π (cid:90) E k ds ≤ ε N π (cid:107) χ (cid:107) ∞ + ε < η . (cid:3) This theorem states that | f ε ( z ) | is an (LS)-approximant of χ ( e is ).Now, since f ε ( z ) is a function in H that doesn’t vanish at the origin,the n -th optimal polynomial approximant q n of 1 /f ε is non-vanishingon D (see, e.g., Theorem 4.1). This suggests that q n (more specifically,the reverse polynomial of q n ) should be a part of our rational function H ( z ). This observation leads to the second stage of the approximation.3.5. Second stage of approximation.
Given an ideal filter χ ( e is ),the first stage of the approximation involved determining the function f ε ( z ). It then followed that χ ( e is ) ≈ LS | f ε ( e is ) | . In the next stage, weapproximate f ε with the magnitude of a rational function.Since f ε is actually an outer function in H , it follows that (cid:107) q n f ε − (cid:107) L → n → ∞ , where q n denotes the n -th opa of 1 /f ε . Therefore,given any η >
0, we can choose N so that (cid:107) q N f ε − (cid:107) L < η. Moreoverfor any M ≥
0, we see that (cid:107) q N f ε − p M (cid:107) L = inf p ∈P M (cid:107) q N f ε − p (cid:107) L < η, where p M denotes the orthogonal projection of q N f ε onto P M . Hence,we have that p M /q N is an (LS)-approximant of f ε . Consequently, wehave that | f ε ( e is ) | ≈ LS | p M ( e is ) | / | q N ( e is ) | . URVEY OF OPTIMAL APPROXIMANTS 15
It’s important to note that it’s computationally efficient to determinethe polynomials p M and q N . We have already seen in Theorem 2.1 thatthe coefficients of q N can be expressed as the solution of a system of N +1 linear equations, each of which are dependent only on the function f ε . The entries of the associated matrix B can be expressed as a Fouriercoefficient of the L function | f ε | , i.e., B jk = 12 π (cid:90) π − π | f ε ( e is ) | e i ( k − j ) s ds. Hence, they can be computed efficiently through any available FFTalgorithm. Furthermore, B is a Gram matrix generated by the vectors { z k f ε } Nk =0 . Since these vectors are linearly independent, it follows that B is invertible. Moreover, since Gram matrices are positive definite,and since B is Hermitian and Toeplitz, we can use any of the fastalgorithms to compute its inverse. We then see that the coefficients of q N , say a , . . . , a N , are given by the expression a a ... a N = B − f ε (0)0...0 . Hence, the coefficients are determined by the first column of B − scaledby f ε (0). On the other hand, the coefficients of p M are given by thefirst M + 1 Fourier coefficients of q N f ε .3.6. Third stage of approximation.
In the first two stages of ap-proximation, we were able to approximate an ideal filter χ ( e is ) withthe magnitude of a rational function. More specifically, χ ( e is ) ≈ LS | f ε ( e is ) | ≈ LS | p M ( e is ) || q N ( e is ) | . Since the ideal filter is assumed to be an even function on [ − π, π ], wehave that χ ( e is ) = χ ( e − is ) ≈ LS | f ε ( e − is ) |≈ LS (cid:12)(cid:12)(cid:12)(cid:12) p M ( e − is ) q N ( e − is ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) p ∗ M ( e is ) q ∗ N ( e is ) (cid:12)(cid:12)(cid:12)(cid:12) , where p ∗ M and q ∗ N are the reverse polynomials of p M and q N , respec-tively. Now, since M is arbitrary, choose M ≤ N and define the rationalfunction H ( z ) = p ∗ M ( z ) q ∗ N ( z ) . (9)Then H ( z ) is a rational function that’s analytic at ∞ . It has theproperty that | H ( e is ) | is an approximation of χ ( e is ). Furthermore,since the zeros of q N are outside of D , and since H ( z ) is analytic at ∞ ,it follows that the poles of H ( z ) are contained in the disk D . Therefore,the expression in (9) gives us our stable filter.This three stage method of approximation stems from the ideas pre-sented in 1982 by Chui and Chan in [13]. The use of opas in filterdesign doesn’t seem to have gone much further in the one variablecase, although some papers later in the 80s and 90s discuss related IIRfilter designs (see, e.g., [10, 11, 14]). It would be interesting to knowwhether opas might have some further applications in signal processingresearch. There do remain some open problems in the several variablecase, as discussed in Section 4.2.4. Zeros of opas
One variable spaces.
Let us now turn to a discussion of zerosof opas in the spaces H ω . According to Theorem 2.4, given f ∈ H ,the n -th opa q n is equal to a multiple of the n -th reverse orthonormalpolynomial ϕ ∗ n . Assuming f (0) (cid:54) = 0 , it is well-known (see, e.g., [18, 28])that ϕ n has all its zeros inside the open unit disk D , and therefore ϕ ∗ n ,and hence q n , whose zeros are reflected across the unit circle, have nozeros inside the closed unit disk. From an engineering perspective, for agiven function f , the produced filter whose poles coincide with the zerosof ϕ n is a stable filter. The authors of [8] studied this phenomenon forthe more general spaces H ω . They considered the following “minimal-zero” problem: what is the infimum of the modulus of any zero ofan optimal polynomial approximant q n for any n and any f ∈ H ω (assuming f (0) (cid:54) = 0)? In other words, can any zero of any opa (for anyfunction f ) penetrate the disk for a given space H ω and if so, how farinside the disk can that zero appear? The authors proved the followingresult (see Theorem 2.1 in [8]). Theorem 4.1 ([8]) . Given f ∈ H ω such that f (0) (cid:54) = 0 , for each n ∈ N , let q n,f denote the n -th opa of /f. Consider the extremal problem ofidentifying M := inf (cid:8) | z | : q n,f ( z ) = 0 for some f ∈ H ω and some n ∈ N (cid:9) . URVEY OF OPTIMAL APPROXIMANTS 17
Then if the sequence of weights { ω k } k ∈ N is non-decreasing, M = 1 andthe infimum is not achieved. On the other hand, if the sequence ofweights is such that there exist k, n ∈ N that satisfy ω k + n +1 < ω k +1 / , then M < and the infimum is a minimum. In particular, for any of the Dirichlet-type spaces D α , if α ≥ H or in the classical Dirichlet space) the infimum is 1, andall the zeros of the optimal approximants lie outside the closed unitdisk. On the other hand, if α < , there exist functions f ∈ D α whoseoptimal polynomial approximants have zeros inside the unit disk. Inparticular, in the Bergman space A , the authors showed (see Theorem5.1 of [8] that M = 2 √ / f ( z ) = − z/ √ . The proof of the existence of an extremal in the case that the weightssatisfy the required inequality in Theorem 4.1 from [8] is difficult andrelies on machinery from the theory of orthogonal polynomials. Below,we give an outline of the argument involved in the easier parts of theproof and only mention the general ideas in the proof of existence ofthe extremal.
Proof.
First note that it is enough to consider n = 1, since if z is azero of some optimal approximant q n ( n >
1) for some function f , thensince (cid:107) q n f − (cid:107) ω = (cid:107) ( z − z ) q n z − z f − (cid:107) ω ,z is a zero of a first order approximant for some function that is amultiple of q n z − z f. In that case, if z is a zero of a first order approximant for somefunction f , it is not hard to see that z = (cid:107) zf (cid:107) ω (cid:104) f, zf (cid:105) ω , and therefore the extremal problem becomes to find M = inf (cid:26) (cid:107) zf (cid:107) ω |(cid:104) f, zf (cid:105) ω | , f ∈ H ω (cid:27) . (10)Using the Cauchy-Schwarz inequality and noticing that if the sequence { ω k } k ∈ N is non-decreasing, then (cid:107) zf (cid:107) ω ≥ (cid:107) f (cid:107) ω gives that for thoseweights, M ≥ . Moreover, the condition on the weights given in theintroduction ensure that M ≤ . Therefore M = 1 in that case, andit is not hard to see that equality cannot hold in the Cauchy-Schwarzinequality, which implies that the infimum is not attained. On the other hand, if the weights satisfy the condition that ω k + n +1 <ω k +1 / , the authors of [8] show that if f ( z ) = z k T n (cid:18) z − z (cid:19) , where T n ( g ) is the n -th Taylor polynomial of a function g , then theratio (cid:107) zf (cid:107) ω |(cid:104) f, zf (cid:105) ω | is strictly less than 1, by explicit computation.The remainder of the proof involves reducing the extremal problem(10) to one that only involves functions f that are polynomials withpositive coefficients and of degree at most N . If Q N is the correspond-ing extremal polyomial, the authors then show that the coefficients ofthese extremal polynomials Q N satisfy a three-term recurrence rela-tion, and are therefore orthogonal polynomials on the real line, whichin turn are connected to Jacobi matrices and associated homogeneouslinear differential equations. The authors show that if J is the Jacobimatrix with entries J ij = (cid:113) ω j ω j +1 if | i − j | = 1 and J ij = 0 otherwise,then M = (cid:107) J (cid:107) , where (cid:107) J (cid:107) is the norm of the matrix J. The authorsidentify the extremal function in terms of certain orthonormal poly-nomials associated with Q N and with the norm of the matrix J (seeCorollary 4.5 of [8]). (cid:3) In general, it is very difficult to calculate M explicitly, and the au-thors investigate this question in detail, in particular in some otherspaces involving integral norms, but in particular, the following ques-tion remains open. Open Question 1.
What is M for the spaces D α when α < , α (cid:54) = − α < , α (cid:54) = − , what is the norm of the Jacobi matrix J defined by J ij = (cid:113) ( j +1) α ( j +2) α if | i − j | = 1 and J ij = 0 otherwise?One may also ask the corresponding question for a fixed integer n > . In addition, there are many interesting questions surrounding theextremal function, which satisfies a differential equation and is equalto the derivative of the reproducing kernel evaluated at a particularpoint of the disk, for certain spaces, which seems to warrant furtherinvestigation (see Section 8 of [8]).
URVEY OF OPTIMAL APPROXIMANTS 19
Another direction of inquiry investigates the limit points of the zerosof the optimal polynomial approximants. Indeed, it turns out that for ageneral class of weighted spaces where the weights are associated withcertain “regular measures”, and for (say) cyclic functions f such that1 /f has a singularity on the unit circle, every point of the unit circleis a limit point of the zeros of the optimal polynomial approximants of1 /f. This is an analogue of a beautiful theorem of Jentzsch that statesthat if an analytic function in the unit disk has radius of convergence1, then every point on the unit circle is a limit point of the zeros of theTaylor polynomials of that function. The proof of the theorem followsthe outline of the original proof of Jentzsch’s theorem, but knowledgeof the precise asymptotic behavior of the orthogonal polynomials forregular measures connected to the opas is required. For details, see[8, Theorem 6.2]. In fact, the authors prove that asymptotically, foreach ε > , the zeros of the n -th opa for such a function f lie in adisk of radius 1 + ε. (See Theorem 6.1 in [8].) However, more detail onthe precise behavior of the zeros is needed. For instance, the followingquestion is open. Open Question 2.
Given a cyclic function f ∈ H ω , and given ζ ∈ T , is there a sector with vertex at ζ that is devoid of zeros of opas? Moregenerally, are there regions or rays that the zeros avoid?Finally, another line of investigation related to zeros involves cyclic-ity. In [5, Theorem 6.1], the authors find a characterization of cyclicityof a function f ∈ H based on the relationship between the zeros ofthe opas and the value of f at the origin. Since cyclic functions in H are outer functions and are well-understood, this characterization isnot that useful for H , but perhaps this idea can be extended to otherspaces. Indeed, one might hope that such a characterization would giveinsight into the Brown and Shields Conjecture. Thus, although vague,the following is open. Open Question 3.
For a given space H ω (or, say for the Bergmanspace A or the classical Dirichlet space D ), is there a characterizationof a cyclic function f that relies on the behavior of the zeros of theopas for 1 /f ?4.2. Several variable spaces and Shanks Conjecture.
When con-sidering Hilbert spaces of analytic functions of several variables, the ze-ros of opas are less well-understood, partly because zeros of functions ofseveral variables are no longer isolated. On the other hand, this topicwas of great interest in the engineering literature of the 70s in connec-tion with filters, as discussed earlier. Two recent papers (see [25, 26]) discuss the several variable situation in detail, so we just mention a fewrelevant items of interest here.For simplicity, let us consider the Dirichlet-type spaces of the bidiskdefined as follows. Let ( α , α ) ∈ R . Given f ( z , z ) = ∞ (cid:88) j =0 ∞ (cid:88) k =0 a j,k z j z k analytic in D , we will say f ∈ D α ,α if (cid:107) f (cid:107) α ,α := ∞ (cid:88) j =0 ∞ (cid:88) k =0 ( j + 1) α ( k + 1) α | a j,k | < ∞ . This makes D α ,α into a reproducing kernel Hilbert space. In order todefine optimal polynomial approximants, as the authors of [25] note,one needs to order the monomials z j z k in some fashion. Let us assumethat we have chosen such an ordering (for instance, the degree lexico-graphic ordering), and let P n be the span of the first n + 1 monomials.In this way, given f ∈ D α ,α , not identically 0, we can define, as be-fore, the n -th optimal polynomial approximant of 1 /f in D α ,α to bethe polynomial q n that minimizes (cid:107) pf − (cid:107) α ,α among all polynomials p ∈ P n .In [2], the authors extended their results from [1] to the case when α = α = α to obtain rates of decay for (cid:107) q n f − (cid:107) α,α for functions f analytic in the closed unit bidisk with no zeros in the bidisk. Theyalso gave examples of polynomials with no zeros on the bidisk that arenot cyclic in D α,α for α > /
2. Cyclic polynomials in these Dirichletspaces of the bidisk (for α = α ) were completely characterized in [6].In [21], the authors characterized cyclic polynomials in the anisotropicDirichlet spaces, that is, the more general case of D α ,α when α and α may be different.In [25], the authors discuss the history of the problem of using opti-mal polynomial approximants to design digital filters of two variables.In this quest, in [27], the authors conjectured that if f is a polynomialthat is zero-free in the bidisk, then optimal polynomial approximantsof 1 /f in the Hardy space of the bidisk would also be zero-free in thebidisk. This became known as Shanks Conjecture, which was disprovedin [17]. A simplified counter-example can be found in [25]. However,the following weaker version of the Shanks conjecture remains open. Open Question 4. [Weak Shanks Conjecture] Let f ( z , z ) be a poly-nomial that doesn’t vanish in D . Then the optimal polynomial ap-proximant of 1 /f in H ( D ) is zero-free in D . URVEY OF OPTIMAL APPROXIMANTS 21
It’s interesting to know that although this result is unknown in H ( D ), it does fail in other function spaces of the bidisk, includingthe Bergman space A ( D ) (see Example 22 of [25]).5. Convergence Results
One set of questions of great interest but with little known so far isabout rate of convergence of opas for a given cyclic function f . Since f is cyclic, we know that (cid:107) q n f − (cid:107) ω approaches 0 at the fastest possiblerate in terms of the degree. What is that rate of convergence, for a given f ? In addition, since norm convergence implies uniform convergenceon compact subsets of the open unit disk, we know that q n f approaches1 pointwise, uniformly on compact subsets of D , but what more can besaid about convergence on the circle? In what follows, we discuss whatis known about these questions and some related open problems.5.1. Norm Convergence.
In [1], the authors studied the rate of decayof (cid:107) q n f − (cid:107) α for certain simple cyclic functions f in the Dirichlet-typespaces D α for α ≤ . Note that for α > , a function is cyclic if andonly if it does not vanish in the closed disk. Also, if f is such that1 /f is analytic in the closed disk, then it is easy to see that the rate ofdecay of (cid:107) q n f − (cid:107) α is exponential. Therefore the question of rate ofdecay is most interesting for α ≤ f such that 1 /f has a singularity on the unit circle. The simplest possible function toconsider then is f ( z ) = 1 − z. For the function 1 − z, as we saw in Example 2.2, an explicit formulafor q n can be obtained, and moreover, the authors of [1] calculated therate of decay of (cid:107) q n f − (cid:107) α and showed that this rate of decay doesnot change for any polynomial whose zeros are outside the unit disk(with at least one zero on the circle). They then extended that resultto any function that admits an analytic extension to the closed unitdisk. More specifically, their result is the following. Theorem 5.1.
Let α ≤ and let f be analytic in the closed unit diskand have zeros outside D . Then for each n ∈ N there exists a constant C independent of n such that (cid:107) q n f − (cid:107) α ≤ (cid:40) C ( n +1) − α for α < C log + ( n +1) for α = 1 . Moreover, if f has a zero on the unit circle, this rate of convergence issharp. In [16], the authors showed that the rate of decay of (cid:107) q n f − (cid:107) ω for f ( z ) = 1 − z is precisely equal to (cid:18) (cid:80) n +1 k =0 1 ωk (cid:19) ; the authors of [25]then exploited that single-variable rate to identify rates of decay forvarious analogues of 1 − z in some of the several variable spaces. In[22], the authors came up with an example of a lacunary series ofthe type f ( z ) = 1 + (cid:80) ∞ k =1 a k z k whose corresponding rate of decay inthe Dirichlet-type spaces D α is slower than the rate in Theorem 5.1.For instance, they construct a lacunary function f in H such that1 / (log n ) ε is a lower bound for the rate of decay of (cid:107) q n f − (cid:107) H .However, for most other functions, the rate of decay of (cid:107) q n f − (cid:107) α isunknown, and thus, the following general problem is essentially openfor most functions. Open Question 5.
Given a function f ∈ D α such that f does nothave an analytic extension to the closed disk, find the rate of decay of (cid:107) q n f − (cid:107) α .In [12], the author considered a slightly different convergence ques-tion in the Hardy space H related to double least squares. Given a polynomial f of degree n , for each k ∈ N , let q k be the k -th optimalpolynomial approximant (in H ) of 1 /f. Then let Q n,k be the n -th op-timal polynomial approximant of 1 /q k . Since we expect q k to be somekind of approximation of 1 /f and Q n,k some kind of approximation of1 /q k , we expect Q n,k to approximate f as k → ∞ . Indeed, this will bethe case (as shown in [12]) for cyclic polynomials f (i.e., ones that haveno zeros in D . ) In fact, Chui showed that the limit of the polynomials Q n,k is a polynomial of degree n that preserves the zeros of f that areoutside D and contains in addition the reflection of the zeros of f thatare inside D . We summarize these two results of Chui in the followingtheorem.
Theorem 5.2 ([12],Theorems 2.1 and 3.1) . Let f ( z ) = p ( z ) · (cid:81) mj =1 ( α j − z ) be a polynomial of degree n where α j ∈ D \ { } and p is a polynomialof degree n − m that has no zeros in D . Let q k be the k -th optimalpolynomial approximant in H of /f, and let Q n,k be the n -th optimalpolynomial approximant of /q k . Then as k → ∞ , (cid:107) Q n,k − ˜ f (cid:107) H → , where ˜ f ( z ) = p ( z ) · m (cid:89) j =1 (1 / ¯ α j − z ) . As a consequence, (cid:107) Q n,k − f (cid:107) H → if and only if f is a cyclic poly-nomial, i.e., has no zeros in D . URVEY OF OPTIMAL APPROXIMANTS 23
Izumino (see [20]) extended Chui’s results to functions f ∈ H ∞ suchthat 1 /f ∈ H ∞ using operator theory methods. He also proved aconjecture that Chui had considered related to double least squaresof lower degree than the degree of the original polynomial f . Morespecifically, he proved the following. Theorem 5.3 (Theorem 3.5 of [20]) . Let f ( z ) = p ( z ) · (cid:81) mj =1 ( α j − z ) , where | α j | = 1 for j = 1 , . . . , m and p ∈ H ∞ is an outer function.Then for each n = 0 , . . . , m − , (cid:107) Q n,k (cid:107) H → as k → ∞ . Notice that the convergence result of Chui’s is a result in a finitedimensional space, the space of polynomials of degree at most n, andtherefore convergence of the double least-squares also happens for in-stance uniformly in the closed disk. In general, though, if f is a cyclicfunction, we know ( q n f − z ) converges to 0 on compact subsets ofthe open unit disk, because of the norm convergence, but what hap-pens on the unit circle? For a “good” function, should we expect that q n ( ζ ) → /f ( ζ ) for ζ ∈ T ? This leads to another series of questions thatconcern pointwise convergence of optimal polynomial approximants onthe unit circle, which we discuss in the next section.5.2. Pointwise convergence.
The boundary behavior of optimal poly-nomial approximants depends heavily on the function f whose inversethey are approximating. In two recent papers ([4, 7]), the authors in-vestigate two distinct phenomena. In [4], the authors show that thereexist many H functions f whose opas on the unit circle can approxi-mate any complex number. This is a kind of universality phenomenon.More precisely, they proved a theorem that implies the following. Theorem 5.4 ([4]) . For any ζ ∈ T , there exists a G δ -dense set of func-tions f in H (or in the classical Dirichlet space) with correspondingoptimal approximants q n such that the set of points { q n ( ζ ) } is dense in C . Thus clearly it is not the case that q n ( ζ ) converges to f ( ζ ) for thatparticular point and for that large set of functions f . However, if f iswell-behaved enough, pointwise convergence of the opas at points onthe circle that are not zeros of f does occur. In fact, even more is true,as seen in the following. Theorem 5.5 ([7]) . Let f be a polynomial with simple zeros that lieoutside the open unit disk, and let q n be the n -th opa of /f in theHardy space H or the Bergman space A . Then − q n f converges to uniformly on compact subsets of D \ Z ( f ) , where Z ( f ) is the set ofzeros of f . The proof of this result is somewhat technical and relies on esti-mates of the coefficients of 1 − q n f which are tractable in the Hardyand the Bergman space. It seems likely based on preliminary workthat the result holds for any polynomial with no zeros in D , but thecomputations become more complicated. It would be interesting toknow whether these results still hold for more general spaces and how“good” the functions have to be in order to get convergence versus theuniversality behavior observed in Theorem 5.4. Thus we conclude withthe following open questions. Open Question 6.
Suppose f is a polynomial with zeros Z ( f ) in thecomplement of D and let q n be the n -th opa of 1 /f in H ω . For whichweights { ω k } k ∈ N is it true that 1 − q n f converges to 0 uniformly oncompact subsets of D \ Z ( f )? Open Question 7.
Does Theorem 5.5 hold for functions other thanpolynomials, and if so, for which ones? On the other hand, can wecharacterize the functions for which the universality behavior as inTheorem 5.4 occurs?
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Department of Mathematics, University of South Florida, 4202 E.Fowler Avenue, Tampa, Florida 33620-5700, USA.
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