aa r X i v : . [ m a t h . F A ] M a r UNIONS OF ARCS FROM FOURIER PARTIAL SUMS
DENNIS COURTNEY
Abstract.
Elementary complex analysis and Hilbert space methods showthat a union of at most n arcs on the circle is uniquely determined by the n th Fourier partial sum of its characteristic function. The endpoints of thearcs can be recovered from the coefficients appearing in the partial sum bysolving two polynomial equations. We let T = { z ∈ C : | z | = 1 } and D = { z ∈ C : | z | < } , and for any subset E of T and integer k we write b E ( k ) = 12 π Z E e − ikt dt for the k th Fourier coefficient of the characteristic function χ E of E . As boundedfunctions with the same sequence of Fourier coefficients agree almost everywhere,any subset E of T is determined up to a set of measure zero by the sequence b E ( k ).If E is known to have additional structure, the entire sequence may not be neededto recover E . Our present subject is a simple yet nontrivial illustration of thisprinciple.An arc is by definition a closed, connected, proper and nonempty subset of T .We declare T along with the empty set to be a “union of 0 arcs.” Theorem 1. If n is a nonnegative integer and E and E are unions of at most n arcs satisfying (1) c E ( k ) = c E ( k ) , ≤ k ≤ n, then E = E . Thus a set E that is known to be a union of at most n arcs can be recovered completely from the n th Fourier partial sum of χ E , regardless of any quantitativesense in which this partial sum fails to approximate χ E . This stands in slightcontrast to the well-known defects of Fourier partial sum approximation of functionswith jump discontinuities, such as the Gibbs phenomenon (see e.g. [4, Chapter 17]).Significantly, the property of the Fourier basis expressed by Theorem 1 is not sharedby other orthonormal systems of functions on T (see § Blaschke products , whose properties we recall in §
1. EachBlaschke product has a nonnegative integer order . In § E b E from the set of finite unions of arcs to the set of Blaschke products withthe property that if E is a union of at most n arcs, then b E has order at most n .This map has the property that if E and E satisfy (1), then b E and b E have Date : November 13, 2018.2000
Mathematics Subject Classification.
Primary: 42A16, 46N99.The author was partially supported by the University of Iowa Department of MathematicsNSF VIGRE grant DMS-0602242.
DENNIS COURTNEY the same n th order Taylor polynomial at 0. To prove Theorem 1 it then suffices tonote, as we do in §
3, how a Blaschke product of order at most n is determined byits n th order Taylor polynomial.With Theorem 1 in hand, one may ask how to recover E from a partial list ofFourier coefficients in an explicit fashion. This is the subject of §
4, where we presentan algorithm for testing whether or not a given tuple of complex numbers takes theform ( b E ( k )) nk =0 for a union E of at most n arcs, and for finding the endpoints ofthese arcs in terms of the Fourier coefficients in this case.Perhaps because of its elementary nature, we have not found Theorem 1 explicitlystated in the literature, although it is known, and the literature abounds with moregeneral theorems on the reconstruction of a function from partial knowledge of itsFourier transform. In [6] it is shown that a function on T that is piecewise constanton a partition of T into m connected pieces may be recovered from its m th Fourierpartial sum. Note that Theorem 1 concludes slightly more from a much strongerhypothesis.The argument we use is known to specialists. The basic idea is to apply aconformal map into the disc and then the classical Caratheodory-Fejer theorem [1].This is by no means the only approach to Theorem 1. It should be contrasted withwhat one may get by viewing (1) as a system of polynomial equations and solvingit directly with algebra.We are indebted to Donald Sarason for many valuable discussions, and to MihalisKolountzakis for drawing our attention to [6].1. Blaschke products
Definition.
A (finite) Blaschke product is a function of the form (2) b ( z ) = λ n Y j =1 z − a j − a j z for some nonnegative integer n , some λ ∈ T , and some a , . . . , a n ∈ D . Thenonnegative integer n is called the order of the Blaschke product. If n = 0 we interpret the empty product as 1. The domain of a Blaschke productis either T , D , or the closure D of D , depending on context. A Blaschke product isevidently a rational function that maps T to itself and has no poles in D (it sufficesto check the case n = 1). It is well known that these properties characterize theBlaschke products. Proposition 1.
If a rational function r maps T to itself and has no poles in D ,then it is a Blaschke product of order equal to the number n of zeros of r in D ,counted according to multiplicity.Proof. We induct on n . If n = 0, then r = q − for some polynomial q ; write q ( z ) = P mk =0 q k z k with q m = 0. As q ( T ) ⊆ T we have q ( z ) − = q ( z ) = q (( z ) − ) = m X k =0 q k z − k = P mk =0 q k z m − k z m , z ∈ T , so this holds for all nonzero z ∈ D . As q has no zeros in D , the extreme right handside has no pole at 0; thus m = 0 and q is constant as desired.If r has n + 1 zeros in D , choose one, a , and note that r ( z ) · ( z − a − az ) − has n zerosin D and maps T to itself. (cid:3) NIONS OF ARCS FROM FOURIER PARTIAL SUMS 3
Definition. If b is a Blaschke product, we let U b = { z ∈ T : Im z ≥ } . If the zeros of a Blaschke product are a , . . . , a n , we calculate from (2) zb ′ ( z ) b ( z ) = n X j =1 − | a j | | z − a j | > , z ∈ T , so the argument of b ( e it ) is strictly increasing in t . The argument principle impliesthat b ( e it ) travels n times counterclockwise around T as t runs from 0 to 2 π . Corollary 1.
A Blaschke product b has order n if and only if U b is a disjoint unionof n arcs. This is the main reason we include T as a “union of 0 arcs.”2. Blaschke products from unions of arcs
Let S = { z ∈ C : 0 ≤ z ≤ } and let φ denote the function φ ( z ) = exp(2 πi ( z − / − πi ( z − / . It is easy to show (see e.g. [2, § III.3]) that φ maps S bijectively onto D \ {± } ,that φ restricts to an analytic bijection of the interior of S with D , that φ mapsthe right boundary line of S onto { z ∈ T : Im z > } , and that φ maps the leftboundary line of S onto { z ∈ T : Im z < } . Proposition 2. If E is a disjoint union of n ≥ arcs and h E is given by (3) h E ( z ) = 12 b E (0) + ∞ X k =1 b E ( k ) z k , z ∈ D , then h E is an analytic map of D into S , and the function D → D given by b E = φ ◦ h E extends uniquely to a Blaschke product D → D of order n satisfying U b E = E . Using the formulas for φ and h E one can show without much work that b E isa rational function; the work in proving Proposition 2 is to establish that b E hasthe mapping properties of Proposition 1, and hence is a Blaschke product, and toprove that U b E = E .To motivate the argument, let us work nonrigorously for a moment. Formallywe have the series expansion(4) χ E ( z ) = X k ∈ Z b E ( k ) z k , z ∈ T , and formal manipulation of the series (3) with z ∈ T then shows that χ E ( z ) = h E ( z ) + h E ( z ) = 2 Re h E ( z ) , z ∈ T . As χ E is { , } valued on T , the maximum principle for harmonic functions thenimplies that h E maps D into S , so b E = φ ◦ h E maps D into D and sends the circleto itself. By Proposition 1 it follows that b E is a Blaschke product; the equality U b E = E comes from the mapping properties of φ on the boundary of S .What makes this argument nonrigorous is that the series (4) does not convergefor all z ∈ T , and to equate χ E with 2 Re h E is to ignore the distinction between DENNIS COURTNEY a discontinuous real valued function on T and a harmonic function on D . To fillin these gaps, we need to use the actual connection between 2 Re h E and χ E — theformer is the Poisson integral of the latter. Proof.
It is easily checked that (3) does define an analytic function on D , e.g.because P ∞ k =1 | b E ( k ) | is convergent. One can then verify the identity2 h E ( z ) = 12 π Z π ze − is − ze − is χ E ( e is ) ds, z ∈ D . (Fix z , expand − ze − is as a power series in z and interchange the sum and theintegral.) Taking real parts it follows that for any r ∈ [0 ,
1) and any t (5) 2 Re h E ( re it ) = 12 π Z π P r ( t − s ) χ E ( e is ) ds, where P r ( t ) = Re (cid:18) re it − re it (cid:19) is the Poisson kernel . It is elementary (see e.g. [2, § X.2]) that for r ∈ [0 ,
1) thefunction P r is nonnegative and satisfies π R π P r ( θ ) dθ = 1; thus (5) implies that2 Re h E ( z ) ∈ [0 ,
1] for all z ∈ D , and h E maps D into S .As r increases to 1, the P r converge uniformly to the zero function on the com-plement of any neighborhood of 0 (see e.g. [2, § X.2]). From (5) we conclude(6) lim r ↑ h E ( rz ) = χ E ( z )at any z ∈ T at which χ E is continuous. We conclude that for any such z the limitlim r ↑ ( φ ◦ h E )( rz ) exists and is in T .We claim that φ ◦ h E is a rational function. In the case n = 0 this is clear.Otherwise, from the definition of φ it suffices to show that exp(2 πih E ) is a rationalfunction, and for this it suffices to treat the case n = 1. In this case there arereal numbers a < b with b − a < π satisfying E = { e it : t ∈ [ a, b ] } , and b E ( k ) = exp( − ikb ) − exp( − ika ) − πik for all k >
0. Let log denote the analytic logarithm definedon C \ { z ∈ C : z ≤ } that is real on the positive real axis and recall thatlog(1 − z ) = − P ∞ k =1 z k k for all z ∈ D . A comparison of power series shows h E ( z ) = b − a π + 12 πi (cid:0) log(1 − e − ib z ) − log(1 − e − ia z ) (cid:1) , z ∈ D , so exp(2 πih E ) = exp( i b − a ) − e − ib z − e − ia z is rational.At this point we know that b E = φ ◦ h E is a rational function mapping D intoitself. From (6) we deduce that b E maps T into itself, so b E is a Blaschke productby Proposition 1. The equality U b E = E then follows from (6). The order of b E is n by Corollary 1. (cid:3) If E and E are two unions of arcs related by (1), it is clear from the definitionthat h E and h E have the same n th order Taylor polynomial at 0. As φ is analyticat 0, the same is true of b E and b E . Corollary 2. If n ≥ and E and E are each unions of at most n arcs satisfying (7) c E ( k ) = c E ( k ) , ≤ k ≤ n, NIONS OF ARCS FROM FOURIER PARTIAL SUMS 5 then there are Blaschke products b and b , each of order at most n , satisfying E j = U b j for j = 1 , and (8) b b ( k ) = b b ( k ) , ≤ k ≤ n. Blaschke products from Toeplitz matrices
Fix a positive integer n for the remainder of this section. Our goal is to showthat Blaschke products b and b having order at most n and satisfying (8) mustbe equal. Let L denote the space of square-integrable functions T → C , with innerproduct h f, g i = 12 π Z π f ( e it ) g ( e it ) dt, f, g ∈ L . (We identify two functions if they agree almost everywhere.)For 0 ≤ k ≤ n we let ζ k denote the function T → C given by z z k . It isimmediate that { ζ k : 0 ≤ k ≤ n } is an orthonormal subset of L . We denote itsspan, the space of analytic polynomials of degree at most n , by P ; we let π : L → P denote the orthogonal projection. Definition. If f : T → C is bounded, T f : P → P denotes the linear map given by T f ξ = π ( f ξ ) , ξ ∈ P. Here f ξ is the pointwise product of f and ξ . If we let k T f k denote the norm of T f regarded as a linear operator on P andwrite k f k ∞ = sup z ∈ T | f ( z ) | , it is clear that k T f k ≤ k f k ∞ for any bounded f . It is also clear that for any such f h T f ζ k , ζ j i = b f ( j − k ) , ≤ j, k ≤ n, so the matrix of T f with respect to the orthonormal basis { ζ k : 0 ≤ k ≤ n } isconstant along its diagonals (it is a Toeplitz matrix ).If f is a Blaschke product, then f is analytic on D , so the matrix of T f is lowertriangular with first column ( b f ( k )) nk =0 . Our hypothesis (8) is thus that T b = T b ,and to deduce that b = b it suffices to show how to recover a Blaschke product b of order at most n from the operator T b it induces on P . Lemma 1. If b is a Blaschke product of order at most n , then k T b k = 1 , and forany nonzero r ∈ P satisfying k T b r k = k r k one has T b r = br . This proof is a special case of the proof of [5, Proposition 5.1].
Proof.
There are nonzero polynomials p and q , each of degree at most n , satisfying b = p/q . Clearly T b q = p , and as b maps T to itself, we have | p ( z ) | = | q ( z ) | for all z ∈ T , so k p k = k q k . We deduce that k T b q k = k q k and thus k T b k ≥
1; since also k T b k ≤ k b k ∞ = 1, we conclude k T b k = 1.If r ∈ P satisfies k T b r k = k r k we have k r k = k T b r k = k π ( br ) k ≤ k br k = Z π | b ( e it ) | | r ( e it ) | dt = k r k , from which k π ( br ) k = k br k and thus π ( br ) = br as desired. (cid:3) DENNIS COURTNEY
Remark 1.
The argument of Lemma 1 can be modified to show that if f is boundedand analytic on D and k f k ∞ = 1 , then k T f k ≤ with equality if and only if f is aBlaschke product of order at most n . With more work, one can prove the rest of theclassical Caratheodory-Fejer theorem: that every lower triangular ( n + 1) × ( n + 1) Toeplitz M satisfying k M k = 1 is of the form T f for such an f . We can now prove Theorem 1.
Proof of Theorem 1.
By Corollary 2 there are Blaschke products b and b of orderat most n satisfying U b j = E j for j = 1 , b b ( k ) = b b ( k ) for 0 ≤ k ≤ n . Thissecond fact implies that T b = T b . By Lemma 1 there is nonzero q ∈ P satisfying k T b q k = k T b q k = k q k and b = T b qq = T b qq = b , so E = U b = U b = E . (cid:3) As the Fourier coefficients of a bounded function are coefficients with respect toan orthonormal basis of the Hilbert space L , one might wonder if Theorem 1 is aspecial case of a simpler result about arbitrary orthonormal bases of L . This is notthe case. There are, for example, orthonormal bases B for L with the propertythat for every finite subset F ⊆ B , there is an arc A with the property that everyelement of F is constant on A . (The basis ( e πit f ( t )) f ∈ H , where H is the Haar basis of L [0 ,
1] constructed in [3, § III.1], has this property.) In this situation,if E ⊆ A and E ′ ⊆ A are any two unions of arcs with the same total measure,one will have h χ E , f i = h χ E ′ , f i for all f ∈ F : any finite collection of coefficientswith respect to B must fail to distiguish infinitely many unions of n arcs from oneanother. 4. An algorithm
Let F denote the map sending a union of at most n arcs E to the tuple ( b E ( k )) nk =0 in C n +1 . Suppose c = ( c k ) nk =0 is given, and we desire to know whether or not c in the range of F . The arguments of the previous sections give us the followingprocedure. (We use the orthonormal basis of § P with ( n + 1) × ( n + 1) matrices.)(1) Calculate the n th Taylor polynomial at 0 for φ ( c + P nk =1 c k z k ), and makeits coefficients the first column of a lower-triangular Toeplitz matrix M .(2) Evaluate k M k .If k M k 6 = 1, then c is not in the range of F .(3) Otherwise k M k = 1 and by the Caratheodory-Fejer theorem (see Remark 1)there is a unique Blaschke product f of order at most n satisfying M = T f .Find F = U f (e.g. by solving f ( z ) = ± n th order Taylor polynomial at 0 for b F .If these coefficients are the first column of M then b F = f and c = F ( F );otherwise c is not in the range of F . Remark 2.
The third step of the algorithm is necessary as the map E b E fromunions of n arcs to Blaschke products of order n is not surjective. One can check,for example, that of the Blaschke products b t ( z ) = z n − t − tz n for real | t | < , all ofwhich satisfy U b t = U b , only b is in the range of E b E . NIONS OF ARCS FROM FOURIER PARTIAL SUMS 7
If we know in advance that c = F ( E ) is in the range of F , this algorithm canrecover E from c in a somewhat explicit fashion. The matrix M constructed from c is T b E ; Lemma 1 implies that if we choose a nonzero q ∈ P satisfying k M q k = k q k ,we will have b E = Mqq . If q is chosen so as to have minimal degree, the polynomials M q and q will have no nontrivial common factors. In this case the degree of q isthe order of b E , and the endpoints of the arcs of E — the solutions to b E ( z ) = 1 and b E ( z ) = −
1— are the roots of the polynomials
M q − q and M q + q . A computer hasno difficulty carrying out this procedure to find the arcs of E to any given precisionfrom the tuple c = F ( E ).As this algorithm involves solving polynomial equations, we cannot expect sym-bolic formulas for these endpoints of the arcs of E in terms of the Fourier coefficients b E ( k ). Formulas for the polynomials M q ± q , however, can be obtained with someeffort. The entries of M are polynomials in exp(2 πi b E (0)), b E (1), . . . , b E ( n ) withcomplex coefficients. As M has norm 1, a vector q will satisfy k M q k = k q k if andonly if q is an eigenvector for the self-adjoint matrix M ∗ M corresponding to theeigenvalue 1; we can find such a q by using Gaussian elimination, for example. Asthe entries of M ∗ M are polynomials in the entries of M and their complex conju-gates, the coefficients of q and M q ± q will be rational functions in exp(2 πi b E (0)), b E (1), . . . , b E ( n ) and their complex conjugates. Cases may arise in computing M q ± q symbolically: in row reducing the symbolic matrix M ∗ M − I , one needs toknow whether or not certain functions of the matrix entries are zero— but explicitformulas can be obtained in every case.We give one example. Suppose that E is a union of at most two arcs, with b E (0), b E (1), and b E (2) given. Write E = exp(2 πi b E (0)) and E k = − πik b E ( k ) for k = 1 , E and the denominatorof a = E E + 2 E − E E − E E E E + E E − E + E , are nonzero, then the starting points of the arcs of E are the solutions z of theequation z − az + (cid:18) E + (1 − E ) aE E (cid:19) = 0 . The endpoints of the arcs of E are given by a similar formula. References [1] Constantin Carath´eodory and Leopold Fej´er, ¨Uber den Zusammenhang der Extremen vonharmonischen Funktionen mit ihren Koeffizienten und ¨uber den Picard-Laudau’schen Satz ,Rend. Circ. Mat. Palermo (1911), 218–239.[2] John B. Conway, Functions of one complex variable , second ed., Graduate Texts in Mathe-matics, vol. 11, Springer-Verlag, New York, 1978.[3] Alfred Haar,
Zur Theorie der orthogonalen Funktionensysteme , Math. Ann. (1910), no. 3,331–371.[4] T. W. K¨orner, Fourier analysis , second ed., Cambridge University Press, Cambridge, 1989.[5] Donald Sarason,
Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. (1967), 179–203.[6] Martin Vetterli, Pina Marziliano, and Thierry Blu, Sampling signals with finite rate of inno-vation , IEEE Trans. Signal Process. (2002), no. 6, 1417–1428. Department of Mathematics, University of Iowa, Iowa City, IA 52242
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