An extremal problem for characteristic functions
Isabelle Chalendar, Stephan Ramon Garcia, William T. Ross, Dan Timotin
AAN EXTREMAL PROBLEM FOR CHARACTERISTICFUNCTIONS
ISABELLE CHALENDAR, STEPHAN RAMON GARCIA, WILLIAM T. ROSS,AND DAN TIMOTIN
Abstract.
Suppose E is a subset of the unit circle T and H ∞ ⊂ L ∞ isthe Hardy subalgebra. We examine the problem of finding the distancefrom the characteristic function of E to z n H ∞ . This admits an alternatedescription as a dual extremal problem. Precise solutions are givenin several important cases. The techniques used involve the theory ofToeplitz and Hankel operators as well as the construction of certainconformal mappings. Introduction
The linear extremal problemΛ( ψ ) := sup F ∈ b ( H ) (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) T ψ ( ζ ) F ( ζ ) dζ (cid:12)(cid:12)(cid:12)(cid:12) , (1.1)where b ( H ) is the unit ball of the classical Hardy space H [8, 16] on theopen unit disk D and ψ is in L ∞ of the unit circle T , has been studied bymany different authors over the last century (see [13] for a brief survey and alist of references). For some historical context, let us mention an early resultdue to Fej´er [10], which says that for any complex numbers c , c , . . . , c n onehas Λ Å c z + · · · + c n z n +1 ã = (cid:107) H (cid:107) , where H is the Hankel matrix (blank entries to be treated as zeros) H = c c c · · · c n c c · · · c n c · · · c n ... . . .c n Mathematics Subject Classification.
Key words and phrases.
Extremal problem, truncated Toeplitz operator, Toeplitz opera-tor, Hankel operator, complex symmetric operator.Second author partially supported by National Science Foundation Grant DMS-1001614.Fourth author partially supported by a grant of the Romanian National Authority forScientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0119. a r X i v : . [ m a t h . C V ] A p r I. CHALENDAR, S.R. GARCIA, W.T. ROSS, AND D. TIMOTIN and (cid:107) H (cid:107) denotes the operator norm of H (i.e., the largest singular valueof H ). In the special case when c j = 1, 0 ≤ j ≤ n , Egerv´ary [9] obtainedexplicit formulae for Λ and for the extremal function F . Fej´er’s result wasgeneralized by Nehari in terms of Hankel operators (see [22], [24, Theorem1.1.1] and (2.7) below). We refer the reader to [8,16,17] for further references.It is also known [4, 13] that for each ψ ∈ L ∞ the supremum in (1.1) isequal to sup f ∈ b ( H ) (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) T ψ ( ζ ) f ( ζ ) dζ (cid:12)(cid:12)(cid:12)(cid:12) , (1.2)which is a quadratic extremal problem posed over the unit ball of the Hardyspace H . For rational ψ there are techniques in [4, 13] which lead not onlyto the supremum in (1.1) but also to the extremal function F for whichthe supremum is attained. For general ψ ∈ L ∞ the supremum is difficultto compute and the extremal function may not exist or, even when it doesexist, it may not be unique.In this paper we discuss the family of extremal problems correspondingto ψ ( z ) = χ E ( z ) z n , (1.3)where E is a Lebesgue measurable subset of T , χ E denotes the characteristicfunction of E , and n ∈ Z . We are thus interested in the quantitiesΛ n ( E ) := sup F ∈ b ( H ) (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) E F ( ζ ) ζ n dζ (cid:12)(cid:12)(cid:12)(cid:12) (1.4)If we note that Λ n ( T ) = sup F ∈ b ( H ) (cid:12)(cid:12)(cid:12) “ F ( n − (cid:12)(cid:12)(cid:12) , one may interpret (1.4) as asking how large can be the contribution of theset E to the ( n − st Fourier coefficient of an H function? .The main results of this paper are as follows. Using Hankel operators anddistribution estimates for harmonic conjugates, we first show in Section 3that if E has Lebesgue measure | E | ∈ (0 , π ) thenΛ n ( E ) = , n ≤ . For n ≥
1, we obtain some general estimates in Section 4. The central partof the paper is contained in Section 5, where we use a conformal mappingargument, along with our Hankel operator techniques, to obtain a formulafor Λ n ( I ) when n ≥ I is an arc of the circle T . It is remarkablethat in the case n = 1 the formula is explicit and may be extended to anymeasurable set E ⊂ T with | E | ∈ (0 , π ] (as shown in Section 7); namely,we have Λ ( E ) = 12 sec (cid:32) π | E | π + | E | (cid:33) . For n ≥ E , one obtains an upper boundfor Λ n ( E ). N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 3
Our work stems, perhaps somewhat surprisingly, from additive numbertheory. As noted in [21, p. 325], an approach to the Goldbach conjectureusing the Hardy-Littlewood circle method requires a bound on the expres-sions (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) m f ( x ) e ( − nx ) dx (cid:12)(cid:12)(cid:12)(cid:12) (1.5)where e ( x ) = exp(2 πix ), f is a certain polynomial in e ( x ), and m is aparticular disjoint union of sub-intervals of [0 , ψ is givenby (1.3), are of some relevance to estimating the quantity in (1.5). A similarapproach, where the exponent 2 replaced by 3 in (1.5), was famously usedby I. M. Vinogradov in his celebrated proof that every sufficiently largeodd integer is the sum of three primes. Needless to say, we do not expectour results to help solve the Goldbach conjecture. We merely point outhow these extremal problems are of interest outside complex analysis andoperator theory. 2. Preliminaries
For p ∈ [1 , ∞ ], we let L p represent the standard Lebesgue spaces onthe circle (with respect to normalized Lebesgue measure), with the integralnorms (cid:107) · (cid:107) p for finite p and with the essential supremum norm (cid:107) · (cid:107) ∞ for L ∞ .Let C := C ( T ) denote the complex valued continuous functions on T withthe supremum norm (cid:107) · (cid:107) ∞ . We let H p denote the classical Hardy spaces and H ∞ denote the space of bounded analytic functions on D . As is standard,we identify H p with a closed subspace of L p via non-tangential boundaryvalues on T . See [8, 16] for a thorough treatment of this.For f ∈ L and n ∈ Z , let (cid:98) f ( n ) := (cid:90) π − π f ( e iθ ) e − inθ dθ π denote n -th the Fourier coefficient of f and let (cid:101) f ( e iθ ) := P.V. (cid:90) π − π f ( e i ( θ − ϕ ) ) cot Å θ ã dϕ π (2.1)denote the harmonic conjugate of f .In what follows, E will be a Lebesgue measurable subset of the unit circle T . We use | E | to denote the (non-normalized) Lebesgue measure on T sothat | E | ∈ [0 , π ], so that | I | coincides with arc length whenever I is an arcon T . We will use E − to denote the closure of E .Just so we are not dealing with trivialities in our extremal problem Λ n ( E ),we first dispose of the endpoint cases | E | = 0 and | E | = 2 π . Indeed | E | = 0 ⇒ Λ n ( E ) = 0 , n ∈ Z , (2.2) | E | = 2 π ⇒ Λ n ( E ) = (cid:40) n ≥
10 if n < . (2.3) I. CHALENDAR, S.R. GARCIA, W.T. ROSS, AND D. TIMOTIN
A natural first step in considering any linear extremal problem is to iden-tify the corresponding dual extremal problem. In this case (see [16] for thedetails), we can use the tools of functional analysis to rephrase the originalextremal problem in (1.1) as the dual extremal problemΛ n ( E ) = dist(¯ z n χ E , H ∞ ) = inf {(cid:107) z n χ E − g (cid:107) ∞ : g ∈ H ∞ } . (2.4)It turns out that the above inf can be replaced by a min [20, p. 146]. Sincedist( z n χ E , H ∞ ) = dist( χ E , z n H ∞ )we see, for a fixed set E ⊂ T , thatΛ n ( E ) ≤ Λ n (cid:48) ( E ) , n ≤ n (cid:48) . (2.5)A quick review of the definition of Λ n shows that it is invariant under rota-tion. In other words,Λ n ( e iθ E ) = Λ n ( E ) , n ∈ Z , θ ∈ [0 , π ] . (2.6)The key to our investigation is the fact that Λ n ( E ) can also be expressed interms of the norm of a certain Hankel operator.2.1. Hankel operators.
The
Hankel operator with symbol ϕ ∈ L ∞ is de-fined to be H ϕ : H → H − , H ϕ f := P − ( ϕf ) , where P − denotes the orthogonal projection from H onto H − := L (cid:9) H .With respect to the orthonormal bases { , z, z , . . . } for H and { ¯ z, ¯ z , ¯ z , . . . } for H − , H ϕ has the (Hankel) matrix representation H ϕ = Ä (cid:98) ϕ ( − j − k − ä ≤ j,k< ∞ . By Nehari’s theorem (see [22] [24, Theorem 1.1.1]), (cid:107) H ϕ (cid:107) = dist( ϕ, H ∞ ) , (2.7)from which we conclude, via (2.4), thatΛ n ( E ) = (cid:107) H ¯ z n χ E (cid:107) , (2.8)where H ¯ z n χ E = Ä” χ E ( n − j − k − ä ≤ j,k< ∞ . We also make use of the formula (cid:107) H ϕ (cid:107) e = dist( ϕ, H ∞ + C ) (2.9)for the essential norm of H ϕ , that is, the distance to the compact operators[24, 1.5.3]. Here H ∞ + C denotes the uniformly closed algebra { h + f : h ∈ H ∞ , f ∈ C} endowed with the supremum norm.The Toeplitz operator T ϕ with symbol ϕ ∈ L ∞ is defined to be T ϕ : H → H , T ϕ f = P + ( ϕf ) , where P + is the orthogonal projection of L onto H .The following two results play a key role. The first one is an immediateconsequence of [24, Thm. 7.5.5] and its proof. N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 5
Lemma 2.10.
Let u ∈ L ∞ . (i) If (cid:107) H u (cid:107) e < (cid:107) H u (cid:107) and (cid:107) u (cid:107) ∞ = dist( u, H ∞ ) , then u has constantabsolute value almost everywhere. (ii) If u has constant absolute value almost everywhere, the Toeplitzoperator T u is Fredholm, and ind T u > , then (cid:107) u (cid:107) ∞ = dist( u, H ∞ ) . The second one is from [7] (see also [24, Cor. 3.1.16]).
Lemma 2.11.
Let u be a unimodular function on T . Then T u is Fredholmif and only if (cid:107) H u (cid:107) e < and (cid:107) H u (cid:107) e < . Two results on harmonic conjugates.
In addition to the preced-ing results on Hankel operators, we also need a few classical results fromharmonic analysis. Recall from (2.4) that (cid:101) f is the harmonic conjugate of f ∈ L . Proofs of the following well-known result of Zygmund can be foundin the standard texts [19, V.D] and [16, Corollary III.2.6]. Lemma 2.12.
For each λ < there is a constant C λ > such that if f isreal valued and (cid:107) f (cid:107) ∞ ≤ π/ then π (cid:90) π − π e λ | (cid:101) f ( e iθ ) | dθ ≤ C λ . (2.13) If f is continuous, then for any µ > there exists a constant C f,µ > suchthat π (cid:90) π − π e µ | (cid:101) f ( e iθ ) | dθ ≤ C f,µ . (2.14)Given a real-valued function f ∈ L ∞ , we apply (2.13) and Markov’sinequality σ ( | g | ≥ (cid:15) ) ≤ (cid:15) (cid:90) | g | dσ, (cid:15) > , for a positive measure σ , to the function πf (cid:107) f (cid:107) ∞ to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ® θ : | (cid:101) f ( e iθ ) | > (cid:107) f (cid:107) ∞ λπ log t ´ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C λ t . If (cid:107) f (cid:107) ∞ < , we may choose λ < α := 2 (cid:107) f (cid:107) ∞ λ < . At this point, after some rewriting, we get (cid:12)(cid:12)(cid:12) { θ : | (cid:101) f ( e iθ ) | > y } (cid:12)(cid:12)(cid:12) ≤ Ce − πyα , (2.15) I. CHALENDAR, S.R. GARCIA, W.T. ROSS, AND D. TIMOTIN where C is a constant which is independent of y >
0. Similarly, if f iscontinuous, we obtain from (2.14) that for any µ > C µ > (cid:12)(cid:12)(cid:12) { θ : | (cid:101) f ( e iθ ) | > y } (cid:12)(cid:12)(cid:12) ≤ C µ e − µy . (2.16)This next result of Stein and Weiss is an exact formula for the distributionfunction for (cid:101) χ E [29]. Lemma 2.17. If E is a Lebesgue measurable subset of T then (cid:12)(cid:12)(cid:12) { θ : | (cid:101) χ E ( e iθ ) | > y } (cid:12)(cid:12)(cid:12) = 4 tan − (cid:32) | E | e πy − e − πy (cid:33) . An essential computation.
Putting this all together, we are nowready to prove the following important lemma which is useful in our analysisof a certain family of Hankel operators. It is likely that this next result isalready well-known as a ‘folk theorem’, although we are unable to find aspecific reference for it.
Lemma 2.18. If E is a Lebesgue measurable subset of T and | E | ∈ (0 , π ) ,then dist( χ E , H ∞ + C ) = . Proof.
First observe that dist( χ E , H ∞ + C ) ≤ since H ∞ + C containsthe constant function . Suppose, toward a contradiction, that there existfunctions h in H ∞ and f ∈ C such that (cid:107) χ E − h − f (cid:107) ∞ < . Writing h = u + i (cid:101) u + ib, where u is real-valued and b is a real constant, the inequality | χ E − u − Re f | + | (cid:101) u + b + Im f | = | χ E − h − f | < holds almost everywhere T whence (cid:107) χ E − u − g (cid:107) ∞ < , where g = Re f . Letting M = (cid:107) (cid:101) u (cid:107) ∞ , it follows that (cid:12)(cid:12)(cid:12) { θ : | (cid:101) χ E ( e iθ ) | > y } (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) { θ : | (cid:101) χ E ( e iθ ) − (cid:101) u ( e iθ ) − (cid:101) g ( e iθ ) | > β ( y − M ) } (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) { θ : | (cid:101) g ( e iθ ) | > (1 − β )( y − M ) } (cid:12)(cid:12)(cid:12) (2.19)for any 0 < β <
1. According to (2.15), the first term on the right handside of (2.19) is majorized by Ce − πβ ( y − M ) α for some α < C >
0. Now choose β < β/α >
1, and then µ > − β ) µ > π . Applying (2.16), it follows that the righthand side of (2.19) decreases in y at least as fast as exp( − aπy ) for some a >
1, while, by Lemma 2.17, the order of decrease of the left hand side is
N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 7 exactly exp( − πy ). This leads to a contradiction if y → ∞ which proves thelemma. (cid:3) Lemma 2.20. If E is a Lebesgue measurable subset of T and | E | ∈ (0 , π ) ,then (cid:107) H z n χ E (cid:107) e = , n ∈ Z . (2.21) Proof.
Since the Hankel matrix H z n χ E is obtained from H χ E by either elim-inating or inserting a finite number of columns, we conclude that (cid:107) H z n χ E (cid:107) e = (cid:107) H χ E (cid:107) e = dist( χ E , H ∞ + C ) = by (2.9) and Lemma 2.18. (cid:3) Remark 2.22. (i) There is a weaker version of Lemma 2.18 in [20, VII.A.2] namelydist( ψ, H ∞ ) = 1 , where ψ = 1 on a subset E ⊂ T (with | E | ∈ (0 , π )) and ψ = − T \ E .(ii) If E is a finite union of arcs, then another proof of Lemma 2.20becomes available. Indeed, recall that if ψ : T → C is piecewisecontinuous, then [2] (see also [24, Thm. 1.5.18]) asserts that (cid:107) H ψ (cid:107) e = max ξ ∈ T | ψ ( ξ + ) − ψ ( ξ − ) | , which immediately yields (2.21).2.4. Truncated Toeplitz operators.
In order to study Λ n ( E ) for n ≥
1, we require a few facts about truncated Toeplitz operators, a class ofoperators whose study was spurred by a seminal paper of Sarason [28] (see[15] for a current survey of the subject). Although much of the following canbe phrased in terms of large truncated Toeplitz matrices [3], the argumentsinvolve reproducing kernels and conjugations which are more natural in thesetting of truncated Toeplitz operators [4, 6, 13].For n ≥
1, a simple computation with Fourier series shows that( z n H ) ⊥ := H (cid:9) z n H is the finite dimensional vector space of polynomials of degree at most n − ψ ∈ L ∞ we consider the corresponding truncated Toeplitz operator A n,ψ : ( z n H ) ⊥ → ( z n H ) ⊥ , A n,ψ f = P n ( ψf ) , where P n is the orthogonal projection of L onto ( z n H ) ⊥ . With respect tothe orthonormal basis { , z, z , · · · , z n − } for ( z n H ) ⊥ the matrix represen-tation of A n,ψ is the Toeplitz matrix ( (cid:98) ψ ( j − k )) ≤ j,k ≤ n − .It is easy to see that the map C n : ( z n H ) ⊥ → ( z n H ) ⊥ , defined in termsof boundary functions by( C n f )( ζ ) = f ( ζ ) ζ n − , ζ ∈ T , I. CHALENDAR, S.R. GARCIA, W.T. ROSS, AND D. TIMOTIN is a conjugate-linear, isometric, involution (i.e., a conjugation ). Viewed asa mapping of functions on D , the conjugation C has the explicit form C n n − (cid:88) j =0 a j z j = n − (cid:88) j =0 a n − − j z j , z ∈ D . Moreover, it is also known that A n,ψ satisfies A ∗ n,ψ = C n A n,ψ C n , (i.e., A n,ψ is a complex symmetric operator [11, 12]). Consequently, (cid:107) A n,ψ (cid:107) = max ¶ |(cid:104) A n,ψ f, C n f (cid:105)| : f ∈ ( z n H ) ⊥ , (cid:107) f (cid:107) = 1 © . See [13] for a proof of the preceding result.3.
Evaluation of Λ n ( E ) for n ≤ n ( E ) exactly for n ≤
0. In this setting,Λ n ( E ) is, to a large extent, independent of n and the set E itself. Much ofthe groundwork for this next result has already been done in Section 2. Theorem 3.1. If E is a Lebesgue measurable subset of T with | E | ∈ (0 , π ) then Λ n ( E ) = , n ≤ . Proof.
By duality, note that for any n ≤ n ( E ) = dist( z − n χ E , H ∞ )= inf {(cid:107) z − n χ E − g (cid:107) ∞ : g ∈ H ∞ }≤ inf {(cid:107) z − n χ E − z − n g (cid:107) ∞ : g ∈ H ∞ } = inf {(cid:107) χ E − g (cid:107) ∞ : g ∈ H ∞ }≤ . The last inequality follows since the constant function g ≡ belongs to H ∞ .To establish the reverse inequality, we observe thatΛ n ( E ) = (cid:13)(cid:13)(cid:13) H z − n χ E (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) H z − n χ E (cid:13)(cid:13)(cid:13) e = by Lemma 2.20. (cid:3) Although Theorem 3.1 is quite definitive, its proof is not constructive. Itis therefore of interest to see if, whenever we are presented with a subset E of T with | E | ∈ (0 , π ) and an (cid:15) >
0, we can explicitly construct a function F in the unit ball of H for which the quantity (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) E F ( ζ ) ζ n dζ (cid:12)(cid:12)(cid:12)(cid:12) comes within (cid:15) of . For general n and E , this is most likely an extremelydifficult problem. However, in the special case where n = 0 and E is a finite N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 9 union of arcs, the following method of S. Ja. Khavinson [17, p. 18] furnishesa relatively explicit sequence of functions for which this occurs.Fix N ≥ E be the disjoint union of N open arcs of T , the j th arc proceeding counterclockwise from a j = e iα j to b j = e iβ j so that0 < β j − α j < π and 0 < N (cid:88) j =1 ( β j − α j ) < . Let χ denote the harmonic extension of χ E : T → C to the open unit disk D and normalize the harmonic conjugate (cid:101) χ of χ by requiring that (cid:101) χ (0) = 0.Next observe that g := exp [ π ( − (cid:101) χ + iχ )]is an outer function which maps D onto the open upper half-plane H andsatisfies arg g = πχ E almost everywhere on T . The function ϕ := g − ig + i is therefore inner and satisfies ϕ ( E ) = H ∩ T [14].Now fix (cid:15) > Ä ± Ä − (cid:15) N ä , ± (cid:0) (cid:15) N (cid:1) ä . Letting Γ denote the boundary of Ω, oriented in the positive sense, we notethat the length of Γ is (cid:96) (Γ) = N . Let ρ : D → Ω be a conformal mappingsuch that H ∩ T is mapped to the portion of Γ running, in the positive sense,from ( − (cid:15) N ,
0) to ( − − (cid:15) N , ψ := ρ ◦ ϕ and note that ψ maps D onto Ω while sending each of the N arcs of E onto the upper half of Γ. Inother words, we have ψ ( a j ) = 1 − (cid:15) N and ψ ( b j ) = − − (cid:15) N , j = 1 , , . . . , N. Now define F ( z ) := 2 πiψ (cid:48) ( z )and get (cid:107) F (cid:107) = 12 π (cid:90) π − π | πiψ (cid:48) ( e it ) | dt = (cid:90) π − π | ψ (cid:48) ( e it ) | dt = N (cid:96) (Γ) = 1along with − (cid:15) = N Å − (cid:15) N ã = N (cid:88) i =1 [ ψ ( b i ) − ψ ( a i )]= N (cid:88) i =1 (cid:90) b i a i ψ (cid:48) ( ζ ) dζ = 12 πi (cid:90) T χ E ( ζ ) (2 πiψ (cid:48) ( ζ )) (cid:124) (cid:123)(cid:122) (cid:125) F ( ζ ) dζ. Remark 3.2.
As noted in [20, VII.A.2] there is a unique solution g tothe H ∞ distance extremal problem dist( χ E , H ∞ ). However, there is nomaximizing F for the extremal problem Λ ( E ). Thus approximate solutions F as in the above Khavinson construction is about the best one can do.4. Estimating Λ n ( E ) for n ≥ n ( E ) for n ≤
0, the situa-tion for Λ n ( E ) with n ≥ Theorem 4.1. If E is a Lebesgue measurable subset of T with | E | ∈ (0 , π ) and n ≥ , then max ¶ , | E | π © ≤ Λ n ( E ) < . (4.2) In particular, it follows that lim | E |→ π Λ n ( E ) = 1 for each fixed n ≥ .Proof. From (2.5) we know thatΛ n ( E ) ≥ Λ ( E ) = . On the other hand, setting F ( z ) = z n − in the definition (1.4) yieldsΛ n ( E ) ≥ | E | π , which establishes the lower bound in (4.2).To get the upper bound, we first observe that Λ n ( E ) ≤ n ( E ) = 1. Since Λ n ( E ) = (cid:107) H ¯ z n χ E (cid:107) , it follows from Lemma 2.20 that (cid:107) H ¯ z n χ E (cid:107) e = < (cid:107) H ¯ z n χ E (cid:107) . In light of the fact, from (2.7), that (cid:107) ¯ z n χ E (cid:107) ∞ = 1 = (cid:107) H ¯ z n χ E (cid:107) = dist(¯ z n χ E , H ∞ ) , it follows from Lemma 2.10 (i) that ¯ z n χ E has unit absolute value almosteverywhere on T , and this is obviously not true. This proves the upperbound in (4.2). (cid:3) For n ≥
1, the lower bound in (4.2) is somewhat crude. The followingresult is more precise and can be used to obtain numerical estimates ofΛ n ( E ). N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 11
Theorem 4.3. If n ≥ , then for any α ∈ [ − π, π ] we have Λ n ( E ) ≥ π (cid:90) π − π χ E ( e it ) F n ( t − α ) dt, (4.4) where F n denotes the Fej´er kernel F n ( x ) = sin ( nx ) n sin ( x ) . Proof.
With n ≥ α ∈ [ − π, π ] fixed, let ξ = e iα and let k ξ ( z ) = 1 √ n n − (cid:88) j =0 ( ξz ) j denote the corresponding normalized reproducing kernel for ( z n H ) ⊥ . Ap-plying [4, Thm. 3.1] and [13, Thm. 1] we obtainΛ n ( E ) = sup F ∈ b ( H ) (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) T F ( ζ ) χ E ( ζ ) ζ n dζ (cid:12)(cid:12)(cid:12)(cid:12) = sup f ∈ b ( H ) (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) T f ( ζ ) χ E ( ζ ) ζ n dζ (cid:12)(cid:12)(cid:12)(cid:12) = sup f ∈ b ( H ) | ¨ χ E f, f ζ n − ∂ |≥ sup f ∈ b (( z n H ) ⊥ ) | ¨ χ E f, f ζ n − ∂ | = sup f ∈ b (( z n H ) ⊥ ) | ¨ χ E f, P n ( f ζ n − ) ∂ | = sup f ∈ b (( z n H ) ⊥ ) | (cid:104) A n,χ E f, C n f (cid:105) | = (cid:107) A n,χ E (cid:107)≥ | (cid:104) A n,χ E k ξ , k ξ (cid:105) |≥ n | ξ | n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) π − π χ E ( e it ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e int − ξ n e it − ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 π (cid:90) π − π χ E ( e it ) sin (cid:0) n ( t − α ) (cid:1) n sin (cid:0) t − α (cid:1) dt = 12 π (cid:90) π − π χ E ( e it ) F n ( t − α ) dt, (4.5)where we have used the fact that χ E and F n are both nonnegative. (cid:3) Remark 4.6.
Although not directly related to our investigations, it is worthnoting that the proof of Theorem 4.3 can be used to obtain the well-knownfact that the norm of the Toeplitz operator T ψ on H is given by (cid:107) T ψ (cid:107) = (cid:107) ψ (cid:107) ∞ . Indeed, computations similar to (4.5) lead to (cid:107) A n,ψ (cid:107) ≥ lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) π − π ψ ( e it ) F n ( t − α ) dt (cid:12)(cid:12)(cid:12)(cid:12) = | ψ ( e iα ) | for almost every α in [ − π, π ]. Since A n,ψ is a compression of T ψ it followsthat (cid:107) ψ (cid:107) ∞ ≥ (cid:107) T ψ (cid:107) ≥ (cid:107) A n,ψ (cid:107) ≥ | ψ ( ξ ) | for almost every ξ in T . Therefore (cid:107) T ψ (cid:107) = (cid:107) ψ (cid:107) ∞ , as claimed.It is easy to see that for any ψ ∈ L ∞ we have lim n →∞ dist( ψ, z n H ∞ ) = (cid:107) ψ (cid:107) ∞ . Indeed, assume dist( ψ, z n H ∞ ) = (cid:107) ψ − z n g n (cid:107) ∞ (as noted in Section 2,the distance is attained). If P ( ψ ) is the Poisson extension of ψ inside D (see,for instance, [16]), thensup z ∈ D | P ( ψ )( z ) − z n g n ( z ) | ≤ (cid:107) ψ − z n g n (cid:107) ∞ . Since (cid:107) g n (cid:107) ≤ (cid:107) ψ (cid:107) , we have z n g n ( z ) → z ∈ D , whencelim n →∞ sup z ∈ D | P ( ψ )( z ) − z n g n ( z ) | ≥ sup z ∈ D | P ( ψ )( z ) | = (cid:107) ψ (cid:107) ∞ . In particular, it follows that lim n →∞ Λ n ( E ) = 1 whenever E is a Lebesguemeasurable subset of T with | E | ∈ (0 , π ]. Under certain circumstances, wecan obtain a better estimate for the speed of this convergence. Proposition 4.7. If E is a Lebesgue measurable subset of T which containsa non-degenerate arc, then − Λ n ( E ) = O ( n ) , n → ∞ . Proof.
Without loss of generality, we may assume that E contains the cir-cular arc I from e − iα to e iα where α ∈ (0 , π ). In light of Theorem 4.3 weconclude that Λ n ( E ) ≥ π (cid:90) α − α F n ( x ) dx = 1 − π (cid:90) α ≤| x |≤ π F n ( x ) dx = 1 − π (cid:90) α ≤| x |≤ π sin ( nx ) n sin ( x ) dx ≥ − απn sin ( α ) . (cid:3) Question 4.8.
Suppose that E is a totally disconnected subset of T whichhas positive measure (for instance, if E is a ‘fat Cantor set’). What can besaid about the rate at which Λ n ( E ) tends to 1? Example 4.9.
We remark that we are free to maximize the lower bound(4.4) with respect to the parameter α ∈ [ − π, π ]. If t ∈ (0 , π ) and E t denotes N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 13 the arc of T from e − it to e it then evaluating the right hand side of (4.4)when α = 0 gives us the integral12 π (cid:90) t − t n Ç sin( nx/ x/ å dx. This integral can be computed directly yielding the following lower estimatesΛ ( E t ) = , Λ ( E t ) ≥ t , Λ ( E t ) ≥ t t ) π , Λ ( E t ) ≥ t t ) + sin(2 t )3 π , Λ ( E t ) ≥ t t ) + sin(2 t ) + sin(3 t )2 π . From the estimate max ¶ , | E t | π © ≤ Λ n ( E t )in (4.2) we observe that the above lower estimates only become meaningfulwhen the right hand sides of the above expressions are greater then (whichwill happen when t is bounded away from 0). As noted in (2.6), the aboveestimates hold for any arc of T with length 2 t .5. The case of the arc
Suppose that α ∈ (0 , π ) and let I α = { e it : t ∈ ( − α, α ) } . We assume n ≥
1. It is known [20, p. 146] that the infimum in (2.4) is attained. Wewill compute the minimizing function, thus obtaining a formula for Λ n ( I α )which is explicit when n = 1. As noted in (2.6), our formula will hold notonly for I α but for any arc of T with length 2 α . Moreover, we will see inSection 7 that for n = 1 it can be extended to any measurable set.5.1. Conformal maps.
We will use the notation ˘ ( ζ, η )for ζ (cid:54) = η ∈ T to denote the sub-arc of T from ζ to η in the positive direction. Remark 5.1.
To avoid confusion later on, it is important to take carefulnote of the direction one traverses the arc ˘ ( ζ, η ). One needs to traverse thisarc from ζ to η always keeping D on the left. For example, ˇ(cid:0) ( e iπ/ , e − iπ/ )is the arc which travels the long way around the circle from e iπ/ to e − iπ/ while ˇ(cid:0) ( e − iπ/ , e iπ/ ) (a) (b) Figure 1. (A) The components (shaded) of the pre-image of theclosed unit disk centered at 4 / z (cid:55)→ z . (B) The region(shaded) O , / . travels the short way around.For fixed n ≥ r ∈ [ ,
1] let O n,r be the domain in C defined by O n,r := D \ ß (cid:12)(cid:12)(cid:12)(cid:12) z n − r (cid:12)(cid:12)(cid:12)(cid:12) ≤ , − π n ≤ arg z ≤ π n ™ . This domain O n,r is obtained as follows. The pre-image of the closed unitdisk centered at r via the mapping z (cid:55)→ z n has n components (see Figure1a). We form O n,r by removing from D the component containing 1 (seeFigure 1b). Then O n,r is a simply connected domain which is symmetricwith respect to R . For n ≥ O n, / = D . Also note that O n,r (cid:48) ⊂ O n,r , r (cid:48) > r. (5.2)We let w ± n,r be the two ‘corners’ of ∂ O n,r characterized by ∂ O n,r ∩ T = ˇ(cid:0) ( w + n,r , w − n,r ) . Lemma 5.3.
Suppose α ∈ (0 , π ) . For every n ≥ there exist an r n,α ∈ ( , and a conformal homeomorphism Φ n,α : D → O n,r n,α such that Φ n,α (0) = 0 and, denoting Φ n,α to also be its continuous extensionto D − , we have Φ n,α ( e ± iα ) = w ± n,r n,α .Proof. Fix n ≥ r ∈ ( , O n,r is simply connected andso there is a unique conformal homeomorphism ϕ r satisfying ϕ r : D → O n,r , ϕ r (0) = 0 , ϕ (cid:48) r (0) > . N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 15 (a) (b) (c)
Figure 2. (A) O ,. , (B) O , . , (C) O , . Since O n,r is symmetric with respect to R , it is easy to see that ϕ r (¯ z ) satisfiesthe same conditions, and thus by uniqueness, we have ϕ r ( z ) = ϕ r (¯ z ). Inparticular, ϕ r (( − , ⊂ R . We also see that ϕ r extends continuously to T and satisfies the conditions ϕ r ( −
1) = − ϕ r (1) = Å − rr ã /n . Finally, again by uniqueness, ϕ / ( z ) = z , i.e., ϕ / is the identity map.If r k , r ∈ [ ,
1) and r k → r , one sees first that the domains O n,r k and O n,r satisfy the hypothesis of the Carath´eodory kernel theorem [25, Theorem1.8] and thus ϕ r k ( z ) → ϕ r ( z ) for all z ∈ D . Then O n,r k and ϕ r k satisfy thehypotheses of [25, Corollary 2.4], whence it follows that ϕ r k → ϕ r uniformlyon T . See Figure 2 for an illustration of the dependence of the domain O n,r on the parameter r .In particular, for our fixed α ∈ (0 , π ), the map from [ ,
1) to C definedby r (cid:55)→ ϕ r ( e iα ) is continuous.Recalling that ˇ(cid:0) ( e iα , e − iα ) goes from e iα to e − iα the long way around T and, similarly, ˇ(cid:0) ( w + n,r , w − n,r ) goes from the upper corner w + n,r to the lowercorner w − n,r the long way around T , suppose that r ∈ [ ,
1) is such that ϕ r ( ˇ(cid:0) ( e iα , e − iα )) ⊂ ˇ(cid:0) ( w + n,r , w − n,r ) . (5.4)Then ϕ r can be continued by Schwarz reflection to a function Φ r analytic in “ C \ ˇ(cid:0) ( w − n,r , w + n,r ) , and the range of any such Φ r (for fixed r > ) does not contain a fixedneighborhood of the point 1.Suppose now that (5.4) is true for every r ∈ [ , r k → r k − , k ≥ , form a normal family in the domainΩ n := “ C \ ˇ(cid:0) ( e − iπ/ n , e iπ/ n ) , the intersection of the decreasing (see (5.2)) domains “ C \ ˇ(cid:0) ( w − n,r nk , w + n,r n,k ) , k ≥ . By passing to a subsequence, we may assume that (Φ r k − − convergesuniformly on compact subsets of Ω n . Thus Φ r k converges uniformly oncompact subsets of Ω n to some analytic function g . Since Φ r (0) = 0 and ϕ r ( −
1) = − r , we must have g (0) = 0 and g ( −
1) = −
1. Thus g is anon-constant analytic function and so g must be open. On the other hand,if ‹ O n,r := O n,r ∪ ß z ∈ C : 1 z ∈ O n,r ™ − , then the image of Φ r k is contained in ‹ O n,r k . We will now derive a contradic-tion and show that g is not an open map. Indeed, the image of g is containedin (cid:92) k ‹ O n,r k = ‹ O n, and this last set contains 0 in its boundary. But since g (0) = 0, we see that g cannot be an open map.It now follows that (5.4) cannot be true for every r ∈ [ , r = we see that ϕ / ( e iα ) = e iα and w ± n, / = 1, so (5.4) is satisfied. Clearly w ± n,r depends continuously on the parameter r . If we define r n,α := sup ¶ r ∈ [ ,
1) : (5.4) is true for any s ∈ [ , r ) © , then ϕ r n,α ( e ± iα ) = w ± n,r . Indeed, by taking a sequence r k (cid:37) r n,α one sees that | ϕ r n,α ( e ± iα ) | = 1.If, say, Arg( ϕ r n,α ( e iα )) > Arg w + n,r , then, by continuity, this would happenfor all r > r n,α in a small neighborhood of r n,α , which is easily seen tocontradict the definition of r n,α . It follows that r n,α and Φ n,α = ϕ r n,α satisfy the requirements of the lemma. (cid:3) Remark 5.5.
The uniqueness of r n,α and Φ n,α subject to the conditions inLemma 5.3 is a consequence of Theorem 5.6 (see below), since it is shownin its proof that r n,α = dist(¯ z n χ I α , H ∞ ), and that Φ n,α is uniquely definedby prescribing values at the three points 0 , e − iα and e iα .5.2. The heart of the matter.
We have now arrived at the main part ofour argument which requires some technical details of Hankel and Toeplitzoperators. Here is our main result.
Theorem 5.6. If α ∈ [0 , π ] , I α = ˇ(cid:0) ( e − iα , e iα ) , and n ≥ , then Λ n ( I α ) = r n,α . N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 17
Figure 3. If n = 1 then ϕ ( ˝(cid:0) ( e iα , e − iα )) ⊂ r ,α T (solid) and ϕ ( ˝(cid:0) ( e − iα , e iα )) ⊂ r ,α T (dashed). Proof.
Fix α and n and let I = I α and define ϕ := r n,α Φ nn,α . From Lemma 5.3 it follows that ϕ is analytic on D , continuous on D − , andsatisfies ϕ ( ˇ(cid:0) ( e iα , e − iα )) ⊂ r n,α T and ϕ ( ˇ(cid:0) ( e − iα , e iα )) ⊂ r n,α T (see Figure 3). Therefore, the function u : T → C defined by u := ¯ z n ( ϕ − χ I )has constant absolute value equal to r n,α . Also observe that ϕ has a zero oforder n at the origin allowing us to write ϕ = z n ϕ and thus u = ϕ − ¯ z n χ I . We want to apply Lemma 2.10 to u . First, u has constant absolute valueon T . Second T u = T ¯ z n T ϕ − χ I (5.7)and so T u is Fredholm if and only if T ϕ − χ I is Fredholm. Third, since ϕ iscontinuous we can use (2.9) and Lemma 2.20 to see that (cid:107) H ϕ − χ I (cid:107) e = (cid:107) H ϕ − χ I (cid:107) e = (cid:107) H χ I (cid:107) e = < r n,α . It now follows from Lemma 2.11 that T ϕ − χ I is Fredholm.To compute the index of T u , note that ϕ − χ I is piecewise continuous,with two discontinuity points at e − iα and e iα . By [27, Theorems 1 and 2] weknow that (i) the harmonic extension P ( ϕ − χ I )( r, t ) of ϕ − χ I is boundedaway from zero in some annulus { z : 1 − (cid:15) < | z | < } ; (ii) for any fixed r ∈ (1 − (cid:15),
1) the curve t (cid:55)→ P ( ϕ − χ I )( r, t ) has the same winding numberwith respect to 0; (iii) the index of T ϕ − χ I is equal to minus this windingnumber. (a) (b) Figure 4. (A) The curve ϕ ( T ) when n = 3. (B) The curve γ corre-sponding to Figure 4a. It has winding number 3 − The compute this winding number, notice that the circles of radius r n,α centered at 0 and 1 intersect at the two points r n,α e ± iβ where cos β = r n,α .We denote by γ the curve obtained by considering the curve ( ϕ − χ I )( T )and then making it into a closed curve by adding, in the appropriate places,the segments[ r n,α e iβ − , r n,α e iβ ] and [ r n,α e − iβ , r n,α e − iβ − n − T u = ind T ¯ z n + ind T ϕ − χ I = n + (1 − n ) = 1 . To finish, Lemma 2.10 (ii) tells us that r n,α = (cid:107) u (cid:107) ∞ = dist( u, H ∞ ) = dist(¯ z n χ I − ϕ , H ∞ )= dist(¯ z n χ I , H ∞ ) = Λ n ( I α )which proves the theorem. (cid:3) Remark 5.8.
Since (cid:107) H ¯ z n χ Iα (cid:107) = r n,α > / (cid:107) H ¯ z n χ Iα (cid:107) e , it follows from a classical result of Adamyan–Arov–Krein (see [24, Theorem1.1.4]) that g = ϕ is the unique minimizing function in (2.4).6. An explicit computation for n = 1When n = 1 one can make explicit computations. In this case O ,r = D \ ß z : (cid:12)(cid:12)(cid:12)(cid:12) z − r (cid:12)(cid:12)(cid:12)(cid:12) ≤ ™ N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 19
Figure 5.
The domain O ,r ,π/ . (see Figure 5) and the corresponding conformal homeomorphisms can becomputed in closed form, leading to a precise formula for Λ ( I α ), where I α = ˇ(cid:0) ( e − iα , e iα ) . Theorem 6.1.
For α ∈ (0 , π ) we have r ,α = 12 sec Å παπ + 2 α ã , Φ ,α ( z ) = Ä z − e iα e iα z − ä ππ +2 α − e iπαπ +2 α Ä e iα z − e iα e iα z − ä ππ +2 α − , where we have taken the principal branch of the power ππ +2 α . Therefore, Λ ( I α ) = 12 sec παπ + 2 α . (6.2) Proof.
Define β := παπ + 2 α . It is easily checked, using the notation in Section 5, that e ± iβ = w ± ,r ,α . If ϕ ( z ) = z − e iα e iα z − , w ( z ) = z β/α − e iβ e iβ z β/α − , then ϕ is a conformal homeomorphism from D to the upper half plane H ,while (see, for instance, [18, page 48]), w is a conformal homeomorphismfrom H to O ,r ,α . We have Φ ,α = w ◦ ϕ , and one can check directly thatΦ ,α (0) = 0 and Φ ,α ( e ± iα ) = e ± iβ . Thus Φ ,α satisfies the conditions ofLemma 5.3. (cid:3) For instance, Λ ( I π/ ) = √ . It also follows from (6.2) that α (cid:55)→ Λ ( I α ) is an increasing function. Question 6.3. Is α (cid:55)→ Λ n ( I α ) an increasing function for every n ? Question 6.4.
When E is an arc and n = 1, is the supremum in (1.4)attained? Remark 6.5.
When n >
1, it does not seem possible to obtain explicitformulas for Φ n,α and r n,α .7. More general sets
The following lemma was proved by Nordgren [23] (see also [5, Theorem7.4.1, Remark 9.4.6]).
Lemma 7.1. If θ is an inner function with θ (0) = 0 , then θ is measurepreserving as a transformation from T to itself. The next result appears in [26, Appendix]. We include the proof forcompleteness.
Theorem 7.2. If E ⊂ T is a measurable set and I ⊂ T is an arc with | I | = | E | , then there exists an inner function θ , with θ (0) = 0 , such that θ − ( I ) and E are equal almost everywhere.Proof. We may assume 0 < | E | < π . Let v be the harmonic extension to D of χ E and by ˜ v its harmonic conjugate. Define S := { z ∈ C : 0 < Re z < } , δ := { z ∈ C : Re z = 0 } , δ := { z ∈ C : Re z = 1 } ; then ψ := v + i ˜ v is ananalytic map from D to S , such that the nontangential limit ψ ( e it ) is almosteverywhere in δ for e it ∈ E and in δ for e it ∈ T \ E .If τ : S → D is the Riemann map that satisfies τ ( ψ (0) = 0, then I := τ ( δ ) and I := τ ( δ ) are complementary arcs on T , while ϕ := τ ◦ ψ isan inner function that satisfies ϕ (0) = 0. Moreover, we have (up to sets ofmeasure 0) E ⊂ ϕ − ( I ) and T \ E ⊂ ϕ − ( I ).Apply Lemma 7.1 to the inner function ϕ to see that | ϕ − ( I ) | = | I | , | ϕ − ( I ) | = | I | , whence | ϕ − ( I ) ∪ ϕ − ( I ) | = 2 π . It follows then that (upto sets of measure 0) E = ϕ − ( I ) and T \ E = ϕ − ( I ). We also have | I | = | E | = | I | and therefore the required inner function θ can be obtainedby composing ϕ with a rotation that maps I onto I . (cid:3) When | ∂E | = 0, an explicit formula for θ may be obtained from [1, Propo-sition 2.1]. Theorem 7.3.
Suppose E ⊂ T , | E | ∈ (0 , π ) . Then for any n ≥ we have Λ n ( E ) ≤ Λ n ( ˇ(cid:0) ( e − i | E | / , e i | E | / )) . Moreover, Λ ( E ) = Λ ( ˇ(cid:0) ( e − i | E | / , e i | E | / )) = r , | E | / . N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 21
Recall that the definition of r n,α is given in Theorem 6.1. Proof.
Let I := ˇ(cid:0) ( e − i | E | / , e i | E | / )and θ be corresponding inner function produced by Theorem 7.2. Then θ ismeasure preserving and χ I ◦ θ = χ E . If g ∈ z n H ∞ , then g ◦ θ ∈ z n H ∞ , andobviously (cid:107) χ I − g (cid:107) ∞ = (cid:107) χ I ◦ θ − g ◦ θ (cid:107) ∞ = (cid:107) χ E − g ◦ θ (cid:107) ∞ . By taking the infimum with respect to all g ∈ z n H ∞ , we obtaindist( χ E , z n H ∞ ) ≤ dist( χ I , z n H ∞ ) , or Λ n ( E ) ≤ Λ n ( I ).To prove the opposite inequality in the case of Λ , note, from the factthat θ is measure preserving, that, if F is in the unit ball of H , then F ◦ θ is also in the unit ball of H . Moreover,12 π (cid:90) π − π χ I F dt = 12 π (cid:90) π − π ( χ I ◦ θ )( F ◦ θ ) dt = 12 π (cid:90) π − π χ E ( F ◦ θ ) dt. By taking the supremum of the absolute value with respect to all F in theunit ball of H , we obtain Λ ( I ) ≤ Λ ( E ), which is the desired inequality. (cid:3) Acknowledgement:
The authors thank Damien Gayet for some usefulsuggestions concerning conformal mappings, and Gilles Pisier for bringingto our attention reference [26].
References
1. Hari Bercovici and Dan Timotin,
Factorizations of analytic self-maps of the upperhalf-plane , Ann. Acad. Sci. Fenn. Math. (2012), no. 2, 649–660. MR 2987092 202. F. F. Bonsall and T. A. Gillespie, Hankel operators with
P C symbols and the space H ∞ + P C , Proc. Roy. Soc. Edinburgh Sect. A (1981), no. 1-2, 17–24. MR 628125(83b:47038) 73. Albrecht B¨ottcher and Bernd Silbermann, Introduction to large truncated Toeplitzmatrices , Universitext, Springer-Verlag, New York, 1999. MR 1724795 (2001b:47043)74. I. Chalendar, E. Fricain, and D. Timotin,
On an extremal problem of Garcia and Ross ,Oper. Matrices (2009), no. 4, 541–546. MR 2597679 (2011b:30130) 2, 7, 115. J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform , MathematicalSurveys and Monographs, vol. 125, American Mathematical Society, Providence, RI,2006. MR 2215991 (2006m:30003) 206. Jeffrey Danciger, Stephan Ramon Garcia, and Mihai Putinar,
Variational principlesfor symmetric bilinear forms , Math. Nachr. (2008), no. 6, 786–802. MR 2418847(2009g:47052) 77. R. G. Douglas and Donald Sarason,
Fredholm Toeplitz operators , Proc. Amer. Math.Soc. (1970), 117–120. MR 0259639 (41 Theory of H p spaces , Academic Press, New York, 1970. 1, 2, 39. E. Egerv´ary, ¨Uber gewisse Extremumprobleme der Funktionentheorie , Math. Ann. (1928), no. 1, 542–561. MR 1512465 2
10. L. Fej´er, ¨Uber gewisse Minimumprobleme der Funktionentheorie , Math. Ann. (1927), no. 1, 104–123. MR 1512357 111. S. R. Garcia and M. Putinar, Complex symmetric operators and applications ,Trans. Amer. Math. Soc. (2006), no. 3, 1285–1315 (electronic). MR 2187654(2006j:47036) 812. ,
Complex symmetric operators and applications. II , Trans. Amer. Math. Soc. (2007), no. 8, 3913–3931 (electronic). MR 2302518 (2008b:47005) 813. S. R. Garcia and W. T. Ross,
A nonlinear extremal problem on the Hardy space , Comp.Methods. Function Theory (2009), no. 2, 485–524. 1, 2, 7, 8, 1114. S. R. Garcia and D. Sarason, Real outer functions , Indiana Univ. Math. J. (2003),no. 6, 1397–1412. MR 2021044 (2004k:30129) 915. Stephan Ramon Garcia and William T. Ross, Recent progress on truncated Toeplitzoperators , Fields Institute Communications (2013), 275–319, http://arxiv.org/abs/1108.1858 . 716. John B. Garnett, Bounded analytic functions , first ed., Graduate Texts in Mathemat-ics, vol. 236, Springer, New York, 2007. MR 2261424 (2007e:30049) 1, 2, 3, 4, 5,1217. S. Ja Khavinson,
Two papers on extremal problems in complex analysis , Amer. Math.Soc. Transl., vol. 129, American Mathematical Society, Providence, 1986. 2, 918. H. Kober,
Dictionary of conformal representations , Dover Publications Inc., NewYork, N. Y., 1952. MR 0049326 (14,156d) 1919. P. Koosis,
Introduction to H p spaces , second ed., Cambridge Tracts in Mathematics,vol. 115, Cambridge University Press, Cambridge, 1998, With two appendices by V.P. Havin [Viktor Petrovich Khavin]. MR 1669574 (2000b:30052) 520. , Introduction to H p spaces , second ed., Cambridge Tracts in Mathematics, vol.115, Cambridge University Press, Cambridge, 1998. MR MR1669574 (2000b:30052)4, 7, 10, 1321. Steven J. Miller and Ramin Takloo-Bighash, An invitation to modern number theory ,Princeton University Press, Princeton, NJ, 2006, With a foreword by Peter Sarnak.MR 2208019 (2006k:11002) 322. Zeev Nehari,
On bounded bilinear forms , Ann. of Math. (2) (1957), 153–162.MR 0082945 (18,633f) 2, 423. Eric A. Nordgren, Composition operators , Canad. J. Math. (1968), 442–449.MR 0223914 (36 Hankel operators and their applications , Springer Monographs inMathematics, Springer-Verlag, New York, 2003. MR 1949210 (2004e:47040) 2, 4, 5,7, 1825. Ch. Pommerenke,
Boundary behaviour of conformal maps , Grundlehren der Math-ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008) 1526. Yanqi Qiu,
On the effect of rearrangement on complex interpolation for families ofBanach spaces , http://arxiv.org/abs/1304.1403 . 20, 2127. Donald Sarason, Toeplitz operators with piecewise quasicontinuous symbols , IndianaUniv. Math. J. (1977), no. 5, 817–838. MR 0463968 (57 Algebraic properties of truncated Toeplitz operators , Oper. Matrices (2007),no. 4, 491–526. MR 2363975 (2008i:47060) 729. E. M. Stein and Guido Weiss, An extension of a theorem of Marcinkiewicz and someof its applications , J. Math. Mech. (1959), 263–284. MR 0107163 (21 N EXTREMAL PROBLEM FOR CHARACTERISTIC FUNCTIONS 23
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