Extremal problems in de Branges spaces: the case of truncated and odd functions
aa r X i v : . [ m a t h . C A ] D ec EXTREMAL PROBLEMS IN DE BRANGES SPACES:THE CASE OF TRUNCATED AND ODD FUNCTIONS
EMANUEL CARNEIRO AND FELIPE GONC¸ ALVES
Abstract.
In this paper we find extremal one-sided approximations of exponential type for a class oftruncated and odd functions with a certain exponential subordination. These approximations optimize the L ( R , | E ( x ) | − d x )-error, where E is an arbitrary Hermite-Biehler entire function of bounded type in theupper half-plane. This extends the work of Holt and Vaaler [22] for the signum function. We also provideperiodic analogues of these results, finding optimal one-sided approximations by trigonometric polynomials ofa given degree to a class of periodic functions with exponential subordination. These extremal trigonometricpolynomials optimize the L ( R / Z , d ϑ )-error, where ϑ is an arbitrary nontrivial measure on R / Z . The periodicresults extend the work of Li and Vaaler [26], who considered this problem for the sawtooth function withrespect to Jacobi measures. Our techniques are based on the theory of reproducing kernel Hilbert spaces(of entire functions and of polynomials) and on the construction of suitable interpolations with nodes at thezeros of Laguerre-P´olya functions. Introduction
Background.
An entire function F : C → C , not identically zero, is said to be of exponential type if τ ( F ) = lim sup | z |→∞ | z | − log | F ( z ) | < ∞ . In this case, the nonnegative number τ ( F ) is called the exponential type of F . We say that F is real entire if F restricted to R is real valued. Given a function f : R → R , a nonnegative Borel measure σ on R , anda parameter δ >
0, we address here the problem of finding a pair of real entire functions L : C → C and M : C → C of exponential type at most δ such that L ( x ) ≤ f ( x ) ≤ M ( x ) (1.1)for all x ∈ R , minimizing the integral Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) d σ ( x ) . (1.2)In the case of Lebesgue measure, this problem dates back to the work of A. Beurling in the 1930’s (see [35]),where the function f ( x ) = sgn( x ) was considered. Further developments of this theory provide the solutionof this extremal problem for a wide class of functions f that includes, for instance, even, odd and truncatedfunctions subject to a certain exponential or Gaussian subordination [7, 8, 10, 11, 12, 14, 19, 27, 28, 29, 35].Several applications of these extremal functions arise in analytic number theory and analysis, for instancein connection to: large sieve inequalities [22, 31, 35], Erd¨os-Tur´an inequalities [11, 20, 26, 35], Hilbert-type Date : September 14, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Extremal functions, de Branges spaces, exponential type, Laplace transform, reproducing kernel,trigonometric polynomials, majorants. nequalities [9, 10, 11, 19, 35], Tauberian theorems [19], inequalities in signal processing [14], and bounds inthe theory of the Riemann zeta-function [4, 5, 6, 13, 15, 18]. Similar approximation problems are treated,for instance, in [16, 17].In the case of general measures σ , the problem (1.1) - (1.2) is still vastly open. In the remarkable paper[22], Holt and Vaaler considered the situation f ( x ) = sgn( x ) and d σ ( x ) = | x | ν +1 d x with ν > −
1. Theysolved this problem (in fact, for a more general class of measures) by establishing an interesting connectionwith the theory of de Branges spaces of entire functions [1]. This idea was further developed in [9] for a classof even functions f with exponential subordination and in [6, 29] for characteristic functions of intervals,both with respect to general de Branges measures. In particular, the optimal construction in [6] was usedto improve the existing bounds for the pair correlation of zeros of the Riemann zeta-function, under theRiemann hypothesis, extending a classical result of Gallagher [15].The purpose of this paper is to complete the framework initiated in [9], where the case of even functionswas treated. Here we develop an analogous extremal theory for a wide class of truncated and odd functions with exponential subordination, with respect to general de Branges measures (these are described below).In particular, this extends the work of Holt and Vaaler [22] for the signum function.1.2. De Branges spaces.
In order to properly state our results, we need to briefly review the main con-cepts and terminology of the theory of Hilbert spaces of entire functions developed by L. de Branges [1].Throughout the text we denote by U = { z ∈ C ; Im ( z ) > } the open upper half-plane. An analytic function F : U → C has bounded type if it can be written as aquotient of two functions that are analytic and bounded in U (or equivalently, if log | F ( z ) | admits a positiveharmonic majorant in U ). If F : U → C is not identically zero and has bounded type, from its Nevanlinnafactorization [1, Theorems 9 and 10], the number v ( F ) = lim sup y →∞ y − log | F ( iy ) | , called the mean type of F , is finite.If E : C → C is entire, we define the entire function E ∗ : C → C by E ∗ ( z ) = E ( z ). A Hermite-Biehler function E : C → C is an entire function that satisfies the basic inequality | E ∗ ( z ) | < | E ( z ) | for all z ∈ U . If E is a Hermite-Biehler function, we define the de Branges space H ( E ) as the space of entirefunctions F : C → C such that k F k E = Z ∞−∞ | F ( x ) | | E ( x ) | − d x < ∞ , (1.3)and such that F/E and F ∗ /E have bounded type in U with nonpositive mean type. This is a Hilbert spacewith inner product given by h F, G i E = Z ∞−∞ F ( x ) G ( x ) | E ( x ) | − d x. The remarkable property about these spaces is that, for each w ∈ C , the evaluation map F F ( w ) is acontinuous linear functional. Therefore, there exists a function K ( w, · ) ∈ H ( E ) such that F ( w ) = h F, K ( w, · ) i E or each F ∈ H ( E ). Such a function K ( w, z ) is called the reproducing kernel of H ( E ).Associated to E , we define the companion functions A ( z ) := 12 (cid:8) E ( z ) + E ∗ ( z ) (cid:9) and B ( z ) := i (cid:8) E ( z ) − E ∗ ( z ) (cid:9) . (1.4)Note that A and B are real entire functions such that E ( z ) = A ( z ) − iB ( z ). The reproducing kernel is givenby [1, Theorem 19] K ( w, z ) = E ( z ) E ∗ ( w ) − E ∗ ( z ) E ( w )2 πi ( w − z ) = B ( z ) A ( w ) − A ( z ) B ( w ) π ( z − w ) , (1.5)and when z = w we have K ( z, z ) = B ′ ( z ) A ( z ) − A ′ ( z ) B ( z ) π . (1.6)From the reproducing kernel property we have K ( w, w ) = h K ( w, · ) , K ( w, · ) i E = k K ( w, · ) k E ≥ , and one can easily show that K ( w, w ) = 0 if and only if w ∈ R and E ( w ) = 0 (see for instance [22, Lemma11] or [1, Problem 45]).1.3. Main results.
For our purposes we let E be a Hermite-Biehler function of bounded type in U . In thiscase, a classical result of M. G. Krein (see [24] or [22, Lemma 9]) guarantees that E has exponential typeand τ ( E ) = v ( E ). Moreover, an entire function F belongs to H ( E ) if and only if it has exponential type atmost τ ( E ) and satisfies (1.3) (see [22, Lemma 12]).Let µ be a (locally finite) signed Borel measure on R satisfying the following two properties:(H1) The measure µ has support bounded by below.(H2) The right-continuous distribution function associated to this measure (that we keep calling µ , witha slight abuse of notation), defined by µ ( x ) := µ (( −∞ , x ]), verifies0 ≤ µ ( x ) ≤ x ∈ R .In some instances we require a third property:(H3) The average value of the distribution function µ is 1, i.e.lim y →∞ y Z y −∞ µ ( x ) d x = 1 . (1.8)We remark that the constant 1 appearing on the right-hand sides of (1.7) and (1.8) could be replaced byany constant C >
0. For simplicity, we normalize the measure (by dilating) to work with C = 1. Observethat any probability measure µ on R satisfying (H1) automatically satisfies (H2) and (H3). Measures liked µ ( λ ) = χ (0 , ∞ ) ( λ ) sin aλ d λ , for a >
0, which were considered by Littmann and Spanier in [30] (giving thetruncated and odd Poisson kernels in the construction below), satisfy (H1) - (H2) but not (H3).Let µ be a signed Borel measure on R satisfying (H1) - (H2). We define the function f µ , the truncatedLaplace transform of this measure, by f µ ( z ) = Z ∞−∞ e − λz d µ ( λ ) , if Re ( z ) > , if Re ( z ) ≤ . (1.9) bserve that f µ is a well-defined analytic function in Re ( z ) > f µ ( z ) = Z ∞−∞ e − λz d µ ( λ ) = Z ∞−∞ ze − λz µ ( λ ) d λ, (1.10)where we have used integration by parts. If we write µ ( − ( y ) := Z y −∞ µ ( x ) d x, under the additional condition (H3) we find that (below we let supp( µ ) ⊂ ( a, ∞ )) f µ (0 + ) = lim x → + f µ ( x ) = lim x → + Z ∞ a xe − λx µ ( λ ) d λ = lim x → + Z ∞ a x e − λx µ ( − ( λ ) d λ = lim x → + Z ∞ ax xe − t µ ( − ( t/x ) d t = Z ∞ te − t d t = 1 , (1.11)by dominated convergence. Our first result is the following. Theorem 1.
Let E be a Hermite-Biehler function of bounded type in U such that E (0) = 0 . Let µ be asigned Borel measure on R satisfying (H1) - (H2) - (H3) . Assume that supp( µ ) ⊂ [ − τ ( E ) , ∞ ) and let f µ bedefined by (1.9) . If L : C → C and M : C → C are real entire functions of exponential type at most τ ( E ) such that L ( x ) ≤ f µ ( x ) ≤ M ( x ) (1.12) for all x ∈ R , then Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | E ( x ) | − d x ≥ K (0 , . (1.13) Moreover, there is a unique pair of real entire functions L µ : C → C and M µ : C → C of exponential type atmost τ ( E ) satisfying (1.12) for which the equality in (1.13) holds. Our second result is the analogous of Theorem 1 for the odd function e f µ ( z ) := f µ ( z ) − f µ ( − z ) . (1.14)Note that if µ is the Dirac delta measure we have e f µ ( x ) = sgn( x ). Theorem 2.
Let E be a Hermite-Biehler function of bounded type in U such that E (0) = 0 . Let µ be asigned Borel measure on R satisfying (H1) - (H2) - (H3) . Assume that supp( µ ) ⊂ [ − τ ( E ) , ∞ ) and let e f µ bedefined by (1.14) . If L : C → C and M : C → C are real entire functions of exponential type at most τ ( E ) such that L ( x ) ≤ e f µ ( x ) ≤ M ( x ) (1.15) for all x ∈ R , then Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | E ( x ) | − d x ≥ K (0 , . (1.16) Moreover, there is a unique pair of real entire functions e L µ : C → C and f M µ : C → C of exponential type atmost τ ( E ) satisfying (1.15) for which the equality in (1.16) holds. Remark 1:
There is no loss of generality in assuming E (0) = 0 and supp( µ ) ⊂ [ − τ ( E ) , ∞ ) in Theorems 1and 2. In fact, since f µ ( x ) and e f µ ( x ) are discontinuous at x = 0, if E (0) = 0 the integrals on the left-handsides of (1.13) and (1.16) always diverge. Given ε >
0, if the set { x ∈ R ; µ ( x ) > } ∩ ( −∞ , − τ ( E ) − ε ) as nonzero Lebesgue measure, we find by (1.10) that f µ ( x ) ≥ C ε x e (2 τ ( E )+ ε ) x for x >
0, and there are noentire functions L and M of exponential type at most 2 τ ( E ) satisfying (1.12) or (1.15). Remark 2:
The minorant problem for f µ can be solved without the hypothesis (H3). We give the detailsin Corollary 7 below. Remark 3:
Note that we are allowing the measure µ to have part of its support on the negative axis.In principle, our function f µ ( x ) could increase exponentially as x → ∞ and does not necessarily be-long to L ( R , | E ( x ) | − d x ) (the same holds for L and M ). When f µ ∈ L ( R , | E ( x ) | − d x ) (resp. e f µ ∈ L ( R , | E ( x ) | − d x )) it is possible to determine the corresponding optimal values of Z ∞−∞ M ( x ) | E ( x ) | − d x and Z ∞−∞ L ( x ) | E ( x ) | − d x separately. This is detailed in Corollaries 8 and 9 below.We use two main tools in the proofs of Theorems 1 and 2. The first is a basic Cauchy-Schwarz inequalityin the Hilbert space H ( E ) that shows that the optimal choice for M ( z ) − L ( z ) must be the square of thereproducing kernel at the origin (divided by a constant). The second tool, used to show the existence of suchoptimal majorants and minorants, is the construction of suitable entire functions that interpolate f µ at thezeros of a given Laguerre-P´olya function. The latter is detailed in Section 2 and extends the construction ofHolt and Vaaler [22, Section 2], that was tailored specifically for the signum function.1.4. A class of homogeneous spaces.
There is a variety of examples of de Branges spaces [1, Chapter 3]for which Theorems 1 and 2 can be directly applied. A basic example would be the classical Paley-Wienerspace H ( E ), when E ( z ) = e − iτz with τ >
0, which gives us extremal functions of exponential type at most2 τ with respect to Lebesgue measure (this extends the classical work of Graham and Vaaler [19]). Anotherinteresting family arises in the discussion of [22, Section 5]. In the terminology of de Branges [1, Section 50],these are examples of homogeneous spaces, and we briefly review their construction below.Let ν > − A ν and B ν given by A ν ( z ) = ∞ X n =0 ( − n (cid:0) z (cid:1) n n !( ν + 1)( ν + 2) . . . ( ν + n ) = Γ( ν + 1) (cid:0) z (cid:1) − ν J ν ( z )and B ν ( z ) = ∞ X n =0 ( − n (cid:0) z (cid:1) n +1 n !( ν + 1)( ν + 2) . . . ( ν + n + 1) = Γ( ν + 1) (cid:0) z (cid:1) − ν J ν +1 ( z ) , where J ν denotes the classical Bessel function of the first kind. If we write E ν ( z ) = A ν ( z ) − iB ν ( z ) , then the function E ν is a Hermite-Biehler function of bounded type in U with exponential type τ ( E ν ) = 1and no real zeros. Observe that when ν = − / A − / ( z ) = cos z and B − / ( z ) = sin z . Fora general ν > −
1, there are positive constants a ν and b ν such that a ν | x | ν +1 ≤ | E ν ( x ) | − ≤ b ν | x | ν +1 (1.17)for all x ∈ R with | x | ≥
1. For each F ∈ H ( E ν ) we have the remarkable identity Z ∞−∞ | F ( x ) | | E ν ( x ) | − d x = c ν Z ∞−∞ | F ( x ) | | x | ν +1 d x , (1.18) ith c ν = π − ν − Γ( ν + 1) − , and from (1.17) and (1.18) we see that F ∈ H ( E ν ) if and only if F hasexponential type at most 1 and either side of (1.18) is finite. Identity (1.18) makes H ( E ν ) the suitable deBranges space to treat the extremal problem (1.1) - (1.2) for the power measure d σ ( x ) = | x | ν +1 d x . Inorder to do so, we define ∆ ν ( δ, µ ) = inf Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | x | ν +1 d x , where the infimum is taken over all pairs of real entire functions L : C → C and M : C → C of exponentialtype at most δ such that L ( x ) ≤ f µ ( x ) ≤ M ( x ) for all x ∈ R . If there is no such a pair we set ∆ ν ( δ, µ ) = ∞ .Define e ∆ ν ( δ, µ ) considering the analogous extremal problem for the odd function e f µ . The following resultfollows from Theorems 1 and 2. Theorem 3.
Let ν > − and δ > . Let µ be a signed Borel measure on R satisfying (H1) - (H2) - (H3) ,and let f µ be defined by (1.9) ( resp. e f µ be defined by (1.14)). We have∆ ν ( δ, µ ) = ( Γ( ν + 1) Γ( ν + 2) (cid:0) δ (cid:1) ν +2 , if supp( µ ) ⊂ [ − δ, ∞ ); ∞ , otherwise; (1.19)and e ∆ ν ( δ, µ ) = ( ν + 1) Γ( ν + 2) (cid:0) δ (cid:1) ν +2 , if supp( µ ) ⊂ [ − δ, ∞ ); ∞ , otherwise . (1.20)If ∆ ν ( δ, µ ) (resp. e ∆ ν ( δ, µ )) is finite, there exists a unique pair of corresponding extremal functions. Proof.
To see why Theorem 3 is indeed a consequence of Theorems 1 and 2 we proceed as follows. For κ >
0, we consider the measure µ κ defined by µ κ (Ω) = µ ( k Ω), where Ω is any Borel measurable set and κ Ω = { κλ ; λ ∈ Ω } . A simple dilation argument shows that∆ ν ( δ, µ ) = κ ν +2 ∆ ν (cid:0) κδ, µ κ − (cid:1) and e ∆ ν ( δ, µ ) = κ ν +2 e ∆ ν (cid:0) κδ, µ κ − (cid:1) , and we can reduce matters to the case δ = 2. Now let L and M be a pair of real entire functions of exponentialtype at most 2 such that L ( x ) ≤ f µ ( x ) ≤ M ( x ) for all x ∈ R , and such that ( M − L ) ∈ L ( R , | x | ν +1 d x ). By(1.17) we have that ( M − L ) ∈ L ( R , | E ν ( x ) | − d x ). Since ( M − L ) is nonnegative on R , according to [22,Theorem 15] (see also [9, Lemma 14]) we can write M ( z ) − L ( z ) = U ( z ) U ∗ ( z ) with U ∈ H ( E ν ). Therefore,by identity (1.18) and Theorem 1, we have Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | x | ν +1 d x = Z ∞−∞ | U ( x ) | | x | ν +1 d x = c − ν Z ∞−∞ | U ( x ) | | E ν ( x ) | − d x = c − ν Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | E ν ( x ) | − d x ≥ c − ν K ν (0 , − , where c ν = π − ν − Γ( ν + 1) − and K ν (0 ,
0) = B ′ ν (0) A ν (0) π = 12 π ( ν + 1) . This establishes (1.19). A similar argument using Theorem 2 gives (1.20). (cid:3)
As illustrated in the argument above, in order to use the general machinery of Theorems 1 and 2 to solvethe extremal problem (1.1) - (1.2) for a given measure σ , one has to first construct an appropriate de Brangesspace H ( E ) that is isometrically contained in L ( R , d σ ). In particular, this construction was carried out in
6] for the measure d σ ( x ) = ( − (cid:18) sin πxπx (cid:19) ) d x, that appears in connection to Montgomery’s formula and the pair correlation of zeros of the Riemann zeta-function (see [32]), and in [30] for the measured σ ( x ) = ( x + a ) d x, where a ≥
0, that appears in connection to extremal problems with prescribed vanishing conditions.1.5.
Periodic analogues.
In Section 4 we consider the periodic version of this extremal problem. Through-out the paper we write e ( z ) = e πiz for z ∈ C . A trigonometric polynomial of degree at most N is an entirefunction of the form W ( z ) = N X k = − N a k e ( kz ) , where a k ∈ C . We say that W is a real trigonometric polynomial if W ( z ) is real for z real. Given a periodicfunction F : R / Z → R , a probability measure ϑ on R / Z and a degree N ∈ Z + , we address in Section 4 theproblem of finding a pair of real trigonometric polynomials L : C → C and M : C → C of degree at most N such that L ( x ) ≤ F ( x ) ≤ M ( x ) (1.21)for all x ∈ R / Z , minimizing the integral Z R / Z (cid:8) M ( x ) − L ( x ) (cid:9) d ϑ ( x ) . (1.22)When ϑ is the Lebesgue measure, this problem was considered, for instance, in [3, 11, 26, 35] in connectionto discrepancy inequalities of Erd¨os-Tur´an type. For general even measures ϑ , the case of even periodicfunctions with exponential subordination was considered in [2, 9]. In [26], Li and Vaaler solved this extremalproblem for the sawtooth function ψ ( x ) = ( x − ⌊ x ⌋ − , if x / ∈ Z ;0 , if x ∈ Z ;with respect to the Jacobi measures. The purpose of Section 4 is to extend the work [26], solving thisproblem for a general class of functions with exponential subordination (which are the periodizations ofour functions f µ and e f µ , including the sawtooth function as a particular case) with respect to arbitrarynontrivial probability measures ϑ (we say that ϑ is trivial if it has support on a finite number of points). Thesolution of this periodic extremal problem is connected to the theory of reproducing kernel Hilbert spaces ofpolynomials and the theory of orthogonal polynomials in the unit circle.2. Interpolation tools
Laplace transforms and Laguerre-P´olya functions.
In this subsection we review some basic factsconcerning Laguerre-P´olya functions and the representation of their inverses as Laplace transforms as in [21,Chapters II to V]. The selected material we need is already well organized in [9, Section 2] and we followclosely their notation. e say that an entire function F : C → C belongs to the Laguerre-P´olya class if it has only real zerosand its Hadamard factorization is given by F ( z ) = F ( r ) (0) r ! z r e − az + bz ∞ Y j =1 (cid:16) − zx j (cid:17) e z/x j , (2.1)where r ∈ Z + , a, b, x j ∈ R , with a ≥ x j = 0 and P ∞ j =1 x − j < ∞ (with the appropriate change ofnotation in case of a finite number of zeros). Such functions are the uniform limits (in compact sets) ofpolynomials with only real zeros. We say that a Laguerre-P´olya function F represented by (2.1) has finitedegree N = N ( F ) when a = 0 and F has exactly N zeros counted with multiplicity. Otherwise we set N ( F ) = ∞ .If F is a Laguerre-P´olya function with N ( F ) ≥
2, and c ∈ R is such that F ( c ) = 0, we henceforth denoteby g c the frequency function given by g c ( t ) = 12 πi Z c + i ∞ c − i ∞ e ts F ( s ) d s. (2.2)Observe that the integral in (2.2) is absolutely convergent since the condition N ( F ) ≥ / | F ( c + iy ) | = O ( | y | − ) as | y | → ∞ . If ( τ , τ ) ⊂ R is the largest open interval containing no zeros of F such that c ∈ ( τ , τ ), the residue theorem implies that g c = g d for any d ∈ ( τ , τ ). Moreover, the Laplacetransform representation 1 F ( z ) = Z ∞−∞ g c ( t ) e − tz d t (2.3)holds in the strip τ < Re ( z ) < τ (the integral in (2.3) is in fact absolutely convergent due to Lemma 4below). If N ( F ) = 0 or 1, we can still represent F ( z ) − as a Laplace transform on vertical strips. In fact, if N ( F ) = 1, we let τ be the zero of F , written in the form (2.1). If τ = 0 then (2.3) holds with g c ( t ) = F ′ (0) − χ ( b, ∞ ) ( t ) , for c > − F ′ (0) − χ ( −∞ ,b ) ( t ) , for c < . (2.4)If τ = 0 then (2.3) holds with g c ( t ) = − τ F (0) − e τ ( t − b ) − χ ( b + τ − , ∞ ) ( t ) , for c > τ ; τ F (0) − e τ ( t − b ) − χ ( −∞ ,b + τ − ) ( t ) , for c < τ. (2.5)If N ( F ) = 0 then (2.3) holds with g c ( t ) = F (0) − δ ( t − b ) , for any c ∈ R , where δ denotes the Dirac delta measure.The fundamental tool for the development of our interpolation theory in this section is the precise qual-itative knowledge of the frequency functions g c . This is extensively discussed in [21, Chapters II to V] andwe collect the relevant facts for our purposes in the next lemma. Lemma 4.
Let F be a Laguerre-P´olya function of degree N ≥ and let g c be defined by (2.2) , where c ∈ R and F ( c ) = 0 . The following propositions hold: (i) The function g c ∈ C N − ( R ) and is real valued. (ii) The function g c is of one sign, and its sign equals the sign of F ( c ) . iii) If ( τ , τ ) ⊂ R is the largest open interval containing no zeros of F such that c ∈ ( τ , τ ) , then forany τ ∈ ( τ , τ ) we have the following estimate (cid:12)(cid:12) g ( n ) c ( t ) (cid:12)(cid:12) ≪ τ,n e τt ∀ t ∈ R , (2.6) where ≤ n ≤ N − .Proof. Parts (i) and (ii) follow from [21, Chapter IV, Theorems 5.1 and 5.3]. Part (iii) follows from [21,Chapter II, Theorem 8.2 and Chapter V, Theorem 2.1]. (cid:3)
Interpolating f µ at the zeros of Laguerre-P´olya functions. In this subsection we constructsuitable entire functions that interpolate our f µ at the zeros of a given Laguerre-P´olya function. In order toaccomplish this, we make use of the representation in (2.3) and Lemma 4. The material in this subsectionextends the classical work of Graham and Vaaler in [19, Section 3], where this construction was achieved forthe particular function F ( x ) = (sin πx ) of Laguerre-P´olya class.If F is a Laguerre-P´olya function, we henceforth denote by α F the smallest positive zero of F (if no suchzero exists, we set α F = ∞ ). Let g = g α F / (if α F = ∞ take g = g ). If µ is a signed Borel measure on R satisfying (H1) - (H2), it is clear that the function g ∗ d µ ( t ) = Z ∞−∞ g ( t − λ ) d µ ( λ ) = Z ∞−∞ g ′ ( t − λ ) µ ( λ ) d λ = g ′ ∗ µ ( t ) (2.7)satisfies the same growth conditions as in (2.6) for τ ∈ (0 , α F ), for 0 ≤ n ≤ N −
3, with the implied constantsnow depending also on µ . We are now in position to define the building blocks of our interpolation. Proposition 5.
Let F be a Laguerre-P´olya function with N ( F ) ≥ . Let g = g α F / and assume that F ( α F / > in case α F = + ∞ , let g = g and assume F (1) > . Let µ be a signed Borel measure on R satisfying (H1) - (H2) , and let f µ be defined by (1.9) . Define A ( F, µ, z ) = F ( z ) Z −∞ g ∗ d µ ( t ) e − tz d t for Re ( z ) < α F , (2.8) A ( F, µ, z ) = f µ ( z ) − F ( z ) Z ∞ g ∗ d µ ( t ) e − tz d t for Re ( z ) > . (2.9) Then z
7→ A ( F, µ, z ) is analytic in Re ( z ) < α F , z
7→ A ( F, µ, z ) is analytic in Re ( z ) > , and thesefunctions are restrictions of an entire function, which we will denote by A ( F, µ, z ) . Moreover, if supp ( µ ) ⊂ [ − τ, ∞ ) , there exists c > so that |A ( F, µ, z ) | ≤ c (cid:0) | z | e τx χ (0 , ∞ ) ( x ) + | F ( z ) | (cid:1) (2.10) for all z = x + iy ∈ C , and A ( F, µ, ξ ) = f µ ( ξ ) (2.11) for all ξ ∈ R with F ( ξ ) = 0 .Proof. We have already noted in (1.10) that z f µ ( z ) is analytic in Re ( z ) > µ satisfies (H1) -(H2). If N ( F ) ≥
3, from (2.7) and Lemma 4 (iii) we see that the integrals on the right-hand sides of (2.8)and (2.9) converge absolutely and define analytic functions in the stated half-planes. If N ( F ) = 2, it can beverified directly that g is continuous and C by parts, and that the function g ′ thus obtained has at most onediscontinuity and still satisfies the growth condition (2.6). Therefore (2.7) holds and, as before, this sufficesto establish the absolute convergence and analiticity of (2.8) and (2.9) in the stated half-planes. ow let 0 < x < α F . Using (2.7), (2.3) and (1.10) we get A ( F, µ, x ) − A ( F, µ, x ) = − f µ ( x ) + F ( x ) Z ∞−∞ g ′ ∗ µ ( t ) e − tx d t = − f µ ( x ) + F ( x ) Z ∞−∞ Z ∞−∞ g ′ ( t − λ ) µ ( λ ) e − tx d λ d t = − f µ ( x ) + F ( x ) Z ∞−∞ (cid:18)Z ∞−∞ g ′ ( t − λ ) e − tx d t (cid:19) µ ( λ ) d λ = − f µ ( x ) + F ( x ) Z ∞−∞ (cid:18)Z ∞−∞ g ′ ( s ) e − sx d s (cid:19) e − λx µ ( λ ) d λ = − f µ ( x ) + Z ∞−∞ x e − λx µ ( λ ) d λ = 0 . This implies that A ( F, µ, z ) = A ( F, µ, z ) in the strip 0 < Re ( z ) < α F . Hence, z
7→ A ( F, µ, z ) and z
7→ A ( F, µ, z ) are analytic continuations of each other and this defines the entire function z
7→ A ( F, λ, z ).The integral representations for A and (2.7) imply, for Re ( z ) ≤ α F /
2, that |A ( F, µ, z ) | ≤ | F ( z ) | Z −∞ | g ′ | ∗ µ ( t ) e − t Re ( z ) d t ≤ | F ( z ) | Z −∞ | g ′ | ∗ µ ( t ) e − t α F / d t , (2.12)while for Re ( z ) ≥ α F / |A ( F, µ, z ) | ≤ | f µ ( z ) | + | F ( z ) | Z ∞ | g ′ | ∗ µ ( t ) e − t α F / d t. (2.13)Since supp( µ ) ⊂ [ − τ, ∞ ) we use (1.10) and (H2) to obtain, for Re ( z ) ≥ α F / | f µ ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞− τ z e − λz µ ( λ ) d λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ | z | (cid:12)(cid:12)(cid:12)(cid:12)Z ∞− τ e − λ Re ( z ) d λ (cid:12)(cid:12)(cid:12)(cid:12) = | z | Re ( z ) e τ Re ( z ) ≤ | z | α F e τ Re ( z ) . (2.14)Estimates (2.12), (2.13) and (2.14) plainly verify (2.10). The remaining identity (2.11) follows from thedefinition of A . (cid:3) Proposition 6.
Let F be a Laguerre-P´olya function that has a double zero at the origin. Let g = g α F / and assume that F ( α F / > in case α F = + ∞ , let g = g and assume F (1) > . Let µ be a signedBorel measure on R satisfying (H1) - (H2) , and let f µ be defined by (1.9) . With z
7→ A ( F, µ, z ) defined byProposition 5, consider the entire functions z L ( F, µ, z ) and z M ( F, µ, z ) defined by L ( F, µ, z ) = A ( F, µ, z ) + g ∗ d µ (0) F ( z ) z (2.15) and M ( F, µ, z ) = L ( F, µ, z ) + 2 F ( z ) F ′′ (0) z . (2.16) The following propositions hold. (i)
We have F ( x ) (cid:8) f µ ( x ) − L ( F, µ, x ) (cid:9) ≥ or all x ∈ R and L ( F, µ, ξ ) = f µ ( ξ ) (2.18) for all ξ ∈ R with F ( ξ ) = 0 . (ii) We have F ( x ) (cid:8) M ( F, µ, x ) − f µ ( x ) (cid:9) ≥ for all x ∈ R and M ( F, µ, ξ ) = f µ ( ξ ) (2.20) for all ξ ∈ R \ { } with F ( ξ ) = 0 . At ξ = 0 we have M ( F, µ,
0) = 1 . (2.21)(iii) The equality (cid:12)(cid:12) M ( F, µ, x ) − f µ ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) f µ ( x ) − L ( F, µ, x ) (cid:12)(cid:12) = 2 | F ( x ) | x F ′′ (0) (2.22) holds for all x ∈ R .Proof. Part (i). For x <
0, using (2.7), (2.8) and (2.15) we get f µ ( x ) − L ( F, µ, x ) = − F ( x ) Z −∞ (cid:8) g ′ ∗ µ ( t ) − g ′ ∗ µ (0) (cid:9) e − tx d t, (2.23)and, for x >
0, using (2.7), (2.9) and (2.15) we get f µ ( x ) − L ( F, µ, x ) = F ( x ) Z ∞ (cid:8) g ′ ∗ µ ( t ) − g ′ ∗ µ (0) (cid:9) e − tx d t. (2.24)If N ( F ) ≥
4, integration by parts in (2.3) shows that the Laplace transforms of g ′ and g ′′ in the strip0 < Re ( z ) < α F are z/F ( z ) and z /F ( z ), respectively (here we use Lemma 4 (iii) to eliminate the boundaryterms). Since F ( α F / >
0, we conclude by Lemma 4 (ii) that g ′ and g ′′ are nonnegative on R . In particular, g ′ is also nondecreasing on R . If N ( F ) = 2 or 3, it can be verified directly that g ′ is nondecreasing on R .In either case, this implies that g ′ ∗ µ is nondecreasing, and (2.17) and (2.18) (for ξ = 0) then follow from(2.23) and (2.24). For ξ = 0 we see directly from (2.11) and (2.15) that L ( F, µ,
0) = 0.
Part (ii). For x <
0, using (2.7), (2.8) and (2.16) we get M ( F, µ, x ) − f µ ( x ) = F ( x ) Z −∞ (cid:26) g ′ ∗ µ ( t ) − g ′ ∗ µ (0) − tF ′′ (0) (cid:27) e − tx d t, (2.25)and, for x >
0, using (2.7), (2.9) and (2.16) we get M ( F, µ, x ) − f µ ( x ) = − F ( x ) Z ∞ (cid:26) g ′ ∗ µ ( t ) − g ′ ∗ µ (0) − tF ′′ (0) (cid:27) e − tx d t. (2.26)In order to prove (2.19) it suffices to verify that (cid:12)(cid:12) g ′ ∗ µ ( t ) − g ′ ∗ µ (0) (cid:12)(cid:12) ≤ | t | F ′′ (0) (2.27)for all t ∈ R .If N ( F ) ≥
4, we have already noted that the Laplace transform of g ′′ in the strip 0 < Re ( z ) < α F is z /F ( z ). Since F ( z ) /z does not vanish at the origin, we see from Lemma 4 that g ′′ ( t ) is nonnegative and ecays exponentially as | t | → ∞ . By a direct verification, the same holds for N ( F ) = 3, where g ′′ mighthave one discontinuity. Thus g ′′ is integrable on R and by (2.3) we find Z ∞−∞ g ′′ ( t ) d t = 2 F ′′ (0) − . (2.28)We are now in position to prove (2.27) for N ( F ) ≥
3. We have already noted in part (i) that g ′ is anondecreasing function. Therefore, for t >
0, we use (H2) and (2.28) to get g ′ ∗ µ ( t ) − g ′ ∗ µ (0) = Z ∞−∞ (cid:8) g ′ ( t − λ ) − g ′ ( − λ ) (cid:9) µ ( λ ) d λ ≤ Z ∞−∞ Z t g ′′ ( s − λ ) d s d λ = 2 t F ′′ (0) − . An analogous argument holds for t <
0. If N ( F ) = 2, we have F ( z ) = F ′′ (0) e bz z and g ( t ) = F ′′ (0) ( t − b ) χ ( b, ∞ ) ( t ), and (2.27) can be verified directly.For ξ = 0, the interpolation property (2.20) follows directly from (2.25) and (2.26). At ξ = 0, since L ( F, µ,
0) = 0, it follows from (2.16) that M ( F, µ,
0) = 1.
Part (iii). Identity (2.22) follows easily from (2.15), (2.16), (2.17) and (2.19). (cid:3) Proofs of the main results
Proof of Theorem 1.
Recall from (1.11) that under (H3) we have f µ (0 + ) = lim x → + f µ ( x ) = 1 . Optimality.
Let L and M be real entire functions of exponential type at most 2 τ ( E ) such that L ( x ) ≤ f µ ( x ) ≤ M ( x ) for all x ∈ R and Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | E ( x ) | − d x < ∞ . Since ( M − L ) is nonnegative on R , by [22, Theorem 15] (or alternatively [9, Lemma 14]) we may write M ( z ) − L ( z ) = U ( z ) U ∗ ( z )with U ∈ H ( E ). Since f µ (0 − ) = 0 and f µ (0 + ) = 1, we find that | U (0) | = M (0) − L (0) ≥
1. From thereproducing kernel identity and the Cauchy-Schwarz inequality, it follows that1 ≤ | U (0) | = (cid:12)(cid:12) h U, K (0 , · ) i E (cid:12)(cid:12) ≤ k U k E k K (0 , · ) k E = k U k E K (0 , , (3.1)and therefore Z ∞−∞ (cid:8) M ( x ) − L ( x ) (cid:9) | E ( x ) | − d x = Z ∞−∞ | U ( x ) | | E ( x ) | − d x = k U k E ≥ K (0 , . (3.2)This establishes (1.13). Moreover, equality in (3.1) (and thus in (3.2)) happens if and only if U ( z ) = c K (0 , z )with | c | = K (0 , − . This implies that we must have M ( z ) − L ( z ) = K (0 , z ) K (0 , . (3.3) .1.2. Existence.
By multiplying E by a complex constant of absolute value 1, we may assume without lossof generality that E (0) ∈ R . Since E is a Hermite-Biehler function of bounded type, we see that E ∗ also hasbounded type. The companion function B defined by (1.4) is then a real entire function of bounded typewith only real zeros. By [1, Problem 34] (see [23] for a generalization) we conclude that B belongs to theLaguerre-P´olya class. The function B has exponential type and it is clear that τ ( B ) ≤ τ ( E ). Note also that B has a simple zero at z = 0 (since E (0) = 0 we have K (0 , > z = 0 cannot be a doublezero of B ).Applying Proposition 6 to the function B ( z ), we construct the entire functions L µ ( z ) = L ( B , µ, z ) (3.4)and M µ ( z ) = M ( B , µ, z ) . (3.5)It follows from (2.17) and (2.19) that L µ ( x ) ≤ f µ ( x ) ≤ M µ ( x )for all x ∈ R . From (2.10), (2.15) and (2.16) if follows that L µ and M µ have exponential type at most 2 τ ( E ).Finally, from (1.5), (1.6), (2.15) and (2.16) we have that M µ ( z ) − L µ ( z ) = B ( z ) B ′ (0) z = K (0 , z ) K (0 , , and as we have seen in (3.3), this is the condition for equality in (1.13).3.1.3. Uniqueness.
From the equality condition (3.3) and the existence of an optimal pair { L µ , M µ } weconclude that this pair must be unique.3.2. Proof of Theorem 2.
Optimality.
This follows as in the optimality part of Theorem 1, just observing that e f µ (0 − ) = − e f µ (0 + ) = 1 . Existence.
We use Proposition 6 with the Laguerre-P´olya functions B ( z ) and its reflection B ( − z )to define e L µ ( z ) = L ( B ( z ) , µ, z ) − M ( B ( − z ) , µ, − z ) (3.6)and f M µ ( z ) = M ( B ( z ) , µ, z ) − L ( B ( − z ) , µ, − z ) . (3.7)These are real entire functions of exponential type at most 2 τ ( E ) that satisfy e L µ ( x ) ≤ e f µ ( x ) ≤ f M µ ( x )for all x ∈ R . As before, from (1.5), (1.6), (2.15) and (2.16) we find that f M µ ( z ) − e L µ ( z ) = 2 B ( z ) B ′ (0) z = 2 K (0 , z ) K (0 , , and this is the condition for equality in (1.16).3.2.3. Uniqueness.
It follows as in the proof of Theorem 1. .3. Further results.
Without assuming (H3) it is possible to solve the minorant problem for f µ . However,we do have to assume that the companion function that generates the nodes of interpolation does not belongto the space H ( E ). Corollary 7.
Let E be a Hermite-Biehler function of bounded type in U such that E (0) > . Let µ be asigned Borel measure on R satisfying (H1) - (H2) . Assume that supp( µ ) ⊂ [ − τ ( E ) , ∞ ) and let f µ be definedby (1.9) . Assume that B / ∈ H ( E ) . Let L µ be the real entire function of exponential type at most τ ( E ) defined by (3.4) . If L : C → C is a real entire function of exponential type at most τ ( E ) such that L ( x ) ≤ f µ ( x ) for all x ∈ R , then Z ∞−∞ (cid:8) f µ ( x ) − L ( x ) (cid:9) | E ( x ) | − d x ≥ Z ∞−∞ (cid:8) f µ ( x ) − L µ ( x ) (cid:9) | E ( x ) | − d x. (3.8) Proof.
From (2.22) and (3.4) we observe first that the right-hand side of (3.8) is indeed finite. If the left-handside of (3.8) is + ∞ there is nothing to prove. Assume then that ( f µ − L ) ∈ L ( R , | E ( x ) | − d x ). We use thefact that there exists a majorant M µ (not necessarily extremal anymore) defined by (3.5), and from (2.22) wesee that ( M µ − f µ ) ∈ L ( R , | E ( x ) | − d x ). By the triangle inequality we get ( M µ − L µ ) ∈ L ( R , | E ( x ) | − d x )and ( M µ − L ) ∈ L ( R , | E ( x ) | − d x ). Since the last two functions are nonnegative on R , from [22, Theorem15] (or alternatively [9, Lemma 14]) we can write M µ ( z ) − L ( z ) = U ( z ) U ∗ ( z )and M µ ( z ) − L µ ( z ) = V ( z ) V ∗ ( z ) , with U, V ∈ H ( E ). This gives us L µ ( z ) − L ( z ) = U ( z ) U ∗ ( z ) − V ( z ) V ∗ ( z ) . Since
B / ∈ H ( E ), from [1, Theorem 22] the set { z E ( ξ ) − K ( ξ, z ); B ( ξ ) = 0 } is an orthogonal basis for H ( E ) (note here that if E ( ξ ) = 0, the function E ( ξ ) − K ( ξ, z ) has to be interpreted as the appropriate limit).We now use Parseval’s identity and the the fact that L µ interpolates f µ at the zeros of B to get Z ∞−∞ (cid:8) L µ ( x ) − L ( x ) (cid:9) | E ( x ) | − d x = Z ∞−∞ (cid:8) | U ( x ) | − | V ( x ) | (cid:9) | E ( x ) | − d x = X B ( ξ )=0 (cid:8) | U ( ξ ) | − | V ( ξ ) | (cid:9) K ( ξ, ξ ) = X B ( ξ )=0 (cid:8) L µ ( ξ ) − L ( ξ ) (cid:9) K ( ξ, ξ ) = X B ( ξ )=0 (cid:8) f µ ( ξ ) − L ( ξ ) (cid:9) K ( ξ, ξ ) ≥ . This concludes the proof of the corollary. (cid:3)
When f µ ∈ L ( R , | E ( x ) | − d x ) it is possible to determine the precise values of the optimal integrals inour extremal problem separately. Corollary 8.
Let E be a Hermite-Biehler function of bounded type in U such that E (0) > . Let µ be asigned Borel measure on R satisfying (H1) - (H2) . Assume that supp( µ ) ⊂ [ − τ ( E ) , ∞ ) and let f µ be defined y (1.9) . Assume that Z ∞−∞ | f µ ( x ) | | E ( x ) | − d x < ∞ (3.9) and that B / ∈ H ( E ) . (i) Let L µ be the extremal minorant of exponential type at most τ ( E ) defined by (3.4) . We have Z ∞−∞ L µ ( x ) | E ( x ) | − d x = X ξ> B ( ξ )=0 f µ ( ξ ) K ( ξ, ξ ) . (3.10)(ii) Assuming (H3) , let M µ be the extremal majorant of exponential type at most τ ( E ) defined by (3.5) .We have Z ∞−∞ M µ ( x ) | E ( x ) | − d x = 1 K (0 ,
0) + X ξ> B ( ξ )=0 f µ ( ξ ) K ( ξ, ξ ) . (3.11) Proof.
We first prove (ii). The function M µ is nonnegative on R and belongs to L ( R , | E ( x ) | − d x ) from(3.9) (observe in particular that E cannot have nonnegative zeros in this situation). From [22, Theorem 15](or alternatively [9, Lemma 14]) we can write M µ ( z ) = U ( z ) U ∗ ( z ) (3.12)with U ∈ H ( E ). We use again the fact that the set { z E ( ξ ) − K ( ξ, z ); B ( ξ ) = 0 } is an orthogonal basisfor H ( E ) since B / ∈ H ( E ) [1, Theorem 22]. From Parseval’s identity and the the fact that M µ interpolates f µ at the zeros of B (with M µ (0) = 1) we arrive at Z ∞−∞ M µ ( x ) | E ( x ) | − d x = Z ∞−∞ | U ( x ) | | E ( x ) | − d x = X B ( ξ )=0 | U ( ξ ) | K ( ξ, ξ ) = X B ( ξ )=0 M µ ( ξ ) K ( ξ, ξ )= 1 K (0 ,
0) + X ξ> B ( ξ )=0 f µ ( ξ ) K ( ξ, ξ ) . This establishes (3.11).We now prove (i). In this case, we still have a majorant M µ (not necessarily extremal anymore) and thefactorization (3.12) still holds. From (2.22) we see that ( M µ − L µ ) ∈ L ( R , | E ( x ) | − d x ) and we can writeagain M µ ( z ) − L µ ( z ) = V ( z ) V ∗ ( z ), with V ∈ H ( E ). This gives us L µ ( z ) = U ( z ) U ∗ ( z ) − V ( z ) V ∗ ( z ) . (3.13)Using Parseval’s identity again, and the fact that L µ interpolates f µ at the zeros of B , we arrive at Z ∞−∞ L µ ( x ) | E ( x ) | − d x = Z ∞−∞ (cid:8) | U ( x ) | − | V ( x ) | (cid:9) | E ( x ) | − d x = X B ( ξ )=0 | U ( ξ ) | − | V ( ξ ) | K ( ξ, ξ )= X B ( ξ )=0 L µ ( ξ ) K ( ξ, ξ ) = X ξ> B ( ξ )=0 f µ ( ξ ) K ( ξ, ξ ) . This establishes (3.10) and completes the proof. (cid:3)
Corollary 9.
Let E be a Hermite-Biehler function of bounded type in U such that E (0) > . Let µ be asigned Borel measure on R satisfying (H1) - (H2) - (H3) . Assume that supp( µ ) ⊂ [ − τ ( E ) , ∞ ) and let e f µ be efined by (1.14) . Assume that Z ∞−∞ | e f µ ( x ) | | E ( x ) | − d x < ∞ (3.14) and that B / ∈ H ( E ) . Let e L µ and f M µ be the extremal functions of exponential type at most τ ( E ) defined by (3.6) and (3.7) , respectively. We have Z ∞−∞ e L µ ( x ) | E ( x ) | − d x = − K (0 ,
0) + X ξ =0 B ( ξ )=0 e f µ ( ξ ) K ( ξ, ξ ) and Z ∞−∞ f M µ ( x ) | E ( x ) | − d x = 1 K (0 ,
0) + X ξ =0 B ( ξ )=0 e f µ ( ξ ) K ( ξ, ξ ) . Proof.
From the integrability condition (3.14) we see that E cannot have real zeros and we may use (3.6),(3.7), (3.12) and (3.13) to write e L µ ( z ) = (cid:0) U ( z ) U ∗ ( z ) − V ( z ) V ∗ ( z ) (cid:1) − U ( z ) U ∗ ( z )and f M µ ( z ) = U ( z ) U ∗ ( z ) − (cid:0) U ( z ) U ∗ ( z ) − V ( z ) V ∗ ( z ) (cid:1) , where U i , V j ∈ H ( E ). Once we have completed this passage from L to L , the remaining steps are analogousto the proof of Corollary 8. (cid:3) Periodic analogues
Recall that we write e ( z ) = e πiz for z ∈ C . In this section we consider the problem of one-sidedapproximation of periodic functions by trigonometric polynomials of a given degree, as described in § Preliminaries.
Reproducing kernel Hilbert spaces of polynomials.
We write D = { z ∈ C ; | z | < } for the open unitdisc and ∂ D for the unit circle. Let n ∈ Z + and let P n be the set of polynomials of degree at most n withcomplex coefficients. If Q ∈ P n we define the conjugate polynomial Q ∗ ,n by Q ∗ ,n ( z ) = z n Q (cid:0) ¯ z − (cid:1) . (4.1)If Q has exact degree n , we sometimes omit the superscript n and write Q ∗ for simplicity.Let P be a polynomial of exact degree n + 1 with no zeros on ∂ D such that | P ∗ ( z ) | < | P ( z ) | (4.2)for all z ∈ D . We consider the Hilbert space H n ( P ) consisting of the elements in P n with scalar product h Q, R i H n ( P ) = Z R / Z Q ( e ( x )) R ( e ( x )) | P ( e ( x )) | − d x. (4.3) rom Cauchy’s integral formula, it follows easily that the reproducing kernel for this finite-dimensionalHilbert space is given by K ( w, z ) = P ( z ) P ( w ) − P ∗ ( z ) P ∗ ( w )1 − ¯ wz , i.e. for every w ∈ C we have the identity h Q, K ( w, · ) i H n ( P ) = Q ( w ) . As before, we define the companion polynomials A ( z ) := 12 (cid:8) P ( z ) + P ∗ ( z ) (cid:9) and B ( z ) := i (cid:8) P ( z ) − P ∗ ( z ) (cid:9) , (4.4)and we find that A = A ∗ , B = B ∗ and P ( z ) = A ( z ) − i B ( z ). Since the coefficients of z and z n +1 of P donot have the same absolute value (this would contradict (4.2) at z = 0) the polynomials A and B have exactdegree n + 1. From (4.2) we also see that A and B have all of their zeros in ∂ D .The reproducing kernel has the alternative representation K ( w, z ) = 2 i B ( z ) A ( w ) − A ( z ) B ( w )1 − ¯ wz ! . (4.5)Observe that K ( w, w ) = h K ( w, · ) , K ( w, · ) i H n ( P ) ≥ w ∈ C . If there is w ∈ C such that K ( w, w ) = 0, then K ( w, · ) ≡ Q ( w ) = 0 for every Q ∈ P n , acontradiction. Therefore K ( w, w ) > w ∈ C . From the representation (4.5) it follows that A and B have only simple zeros and their zeros never agree.From (4.5) we see that the sets { z K ( ζ, z ); A ( ζ ) = 0 } and { z K ( ζ, z ); B ( ζ ) = 0 } are orthogonalbases for H n ( P ) and, in particular, we arrive at Parseval’s formula (see [25, Theorem 2]) || Q || H n ( P ) = X A ( ζ )=0 | Q ( ζ ) | K ( ζ, ζ ) = X B ( ζ )=0 | Q ( ζ ) | K ( ζ, ζ ) . (4.6)4.1.2. Orthogonal polynomials in the unit circle.
The map x e ( x ) allows us to identify measures on R / Z with measures on the unit circle ∂ D . Let ϑ be a nontrivial probability measure on R / Z ∼ ∂ D (recall that ϑ is trivial if it has support on a finite number of points) and consider the space L ( ∂ D , d ϑ ) with inner productgiven by h f, g i L ( ∂ D , d ϑ ) = Z ∂ D f ( z ) g ( z ) d ϑ ( z ) = Z R / Z f ( e ( x )) g ( e ( x )) d ϑ ( x ) . We define the monic orthogonal polynomials Φ n ( z ) = Φ n ( z ; d ϑ ) by the conditionsΦ n ( z ) = z n + lower order terms ; h Φ n , z j i L ( ∂ D , d ϑ ) = 0 (0 ≤ j < n );and we define the orthonormal polynomials by ϕ n = c n Φ n / || Φ n || , where c n is a complex number of absolutevalue one such that ϕ n (1) ∈ R (this normalization will be used later). Observe that h Q ∗ ,n , R ∗ ,n i L ( ∂ D , d ϑ ) = h R, Q i L ( ∂ D , d ϑ ) (4.7)for all polynomials Q, R ∈ P n , where the conjugation map ∗ was defined in (4.1). The next lemma collectsthe relevant facts for our purposes from B. Simon’s survey article [34]. Lemma 10.
Let ϑ be a nontrivial probability measure on R / Z . i) ϕ n has all its zeros in D and ϕ ∗ n has all its zeros in C \ D . (ii) Define a new measure ϑ n on R / Z by d ϑ n ( x ) = d x (cid:12)(cid:12) ϕ n ( e ( x ); d ϑ ) (cid:12)(cid:12) , Then ϑ n is a probability measure on R / Z , ϕ j ( z ; d ϑ ) = ϕ j ( z ; d ϑ n ) for j = 0 , , . . . , n and for all Q, R ∈ P n we have h Q, R i L ( ∂ D , d ϑ ) = h Q, R i L ( ∂ D , d ϑ n ) . (4.8) Proof. (i) This is [34, Theorem 4.1].(ii) This follows from [34, Theorem 2.4, Proposition 4.2 and Theorem 4.3]. (cid:3)
Let n ≥ ϕ n +1 ( z ) = ϕ n +1 ( z ; d ϑ ). By Lemma 10 (i) and the maximum principle we have | ϕ n +1 ( z ) | < | ϕ ∗ n +1 ( z ) | for all z ∈ D . By Lemma 10 (ii) we note (Christoffel-Darboux formula) that P n with the scalar product h· , ·i L ( ∂ D , d ϑ ) is a reproducing kernel Hilbert space with reproducing kernel given by K n ( w, z ) = ϕ ∗ n +1 ( z ) ϕ ∗ n +1 ( w ) − ϕ n +1 ( z ) ϕ n +1 ( w )1 − ¯ wz . (4.9)Observe that ϕ ∗ n +1 plays the role of P in § A n +1 ( z ) = 12 (cid:8) ϕ ∗ n +1 ( z ) + ϕ n +1 ( z ) (cid:9) and B n +1 ( z ) = i (cid:8) ϕ ∗ n +1 ( z ) − ϕ n +1 ( z ) (cid:9) , (4.10)and we note that (4.6) holds.We now derive the quadrature formula that is suitable for our purposes. This result appears in [9,Corollary 26] and we present a short proof here for convenience. Proposition 11.
Let ϑ be a nontrivial probability measure on R / Z and let W : C → C be a trigonometricpolynomial of degree at most N . Let ϕ N +1 ( z ) = ϕ N +1 ( z ; d ϑ ) be the ( N + 1) -th orthonormal polynomial inthe unit circle with respect to this measure and consider K N , A N +1 and B N +1 as defined in (4.9) and (4.10) .Then we have Z R / Z W ( x ) d ϑ ( x ) = X ξ ∈ R / Z A N +1 ( e ( ξ ))=0 W ( ξ ) K N ( e ( ξ ) , e ( ξ )) = X ξ ∈ R / Z B N +1 ( e ( ξ ))=0 W ( ξ ) K N ( e ( ξ ) , e ( ξ )) . Proof.
Write W ( z ) = N X k = − N a k e ( kz )and assume first that W is real valued on R , i.e. a k = a − k . Let τ = min x ∈ R W ( x ) . Then z W ( z ) − τ isa real trigonometric polynomial of degree at most N that is nonnegative on R . By the Riesz-F´ejer theoremthere exists a polynomial Q ∈ P N such that W ( z ) − τ = Q ( e ( z )) Q ( e ( z )) or all z ∈ C . Writing τ = | τ | − | τ | , and using (4.8) and (4.6), we obtain Z R / Z W ( x ) d ϑ ( x ) = Z R / Z (cid:8) | Q ( e ( x )) | + | τ | − | τ | (cid:9) d ϑ ( x )= Z R / Z | Q ( e ( x )) | + | τ | − | τ | (cid:12)(cid:12) ϕ N +1 ( e ( x )) (cid:12)(cid:12) d x = X ξ ∈ R / Z B N +1 ( e ( ξ ))=0 | Q ( e ( ξ )) | + | τ | − | τ | K N ( e ( ξ ) , e ( ξ ))= X ξ ∈ R / Z B N +1 ( e ( ξ ))=0 W ( ξ ) K N ( e ( ξ ) , e ( ξ )) , and analogously at the nodes given by the roots of A N +1 . The general case follows by writing W ( z ) = W ( z ) − i W ( z ), with W ( z ) = P Nk = − N b k e ( kz ) and W ( z ) = P Nk = − N c k e ( kz ), where b k = ( a k + a − k ) and c k = i ( a k − a − k ). (cid:3) Extremal trigonometric polynomials.
We now present the solution of the extremal problem (1.21)- (1.22) for a class of periodic functions with a certain exponential subordination. As described below, thisclass comes from the periodization of the functions f µ and e f µ defined in (1.9) and (1.14).4.2.1. Defining the periodic analogues.
Throughout this section we let µ be a (locally finite) signed Borelmeasure on R satisfying conditions (H1’) - (H2). The condition (H1’) is simply a restriction of our current(H1), namely:(H1’) The measure µ has support on [0 , ∞ ).When convenient, we may require additional properties on µ . The first one is our usual (H3), and we nowintroduce the following summability condition:(H4) The distribution function µ ( x ) := µ (( −∞ , x ]) verifies Z ∞ λ µ ( λ ) d λ < ∞ . For λ > v ( λ, x ) = ( xe − λx if x > , if x ≤ , and define the 1-periodic function h ( λ, x ) := X n ∈ Z v ( λ, x + n ) = e − λ ( x −⌊ x ⌋− ) (cid:8) λ/ x − ⌊ x ⌋ − ) + cosh( λ/ (cid:9) λ/ . If µ is a signed Borel measure satisfying (H1’) - (H2) - (H4) we define the 1-periodic function F µ ( x ) := Z ∞ h ( λ, x ) µ ( λ ) d λ = X n ∈ Z f µ ( x + n ) , (4.11)where the last equality follows from (1.10) and Fubini’s theorem. We observe that F µ is differentiable for x / ∈ Z and that F µ (0 − ) = F µ (0) . or 0 ≤ x ≤ h ( λ, x ) = xe − λx + e − λ (cid:0) − e − λ (cid:1) (cid:16) xe − λx (cid:0) − e − λ (cid:1) + e − λx (cid:17) , and we see from dominated convergence and the computation in (1.11) thatlim sup x → + F µ ( x ) ≤ F µ (0) + 1 , and under the additional condition (H3) we have F µ (0 + ) = F µ (0 − ) + 1 = F µ (0) + 1 . (4.12)We now define the odd counterpart. First we let, for λ > e v ( λ, x ) := v ( λ, x ) − v ( λ, − x )and consider the 1-periodic function e h ( λ, x ) := X n ∈ Z e v ( λ, x + n )= − cosh( λ/
2) sinh (cid:0) λ ( x − ⌊ x ⌋ − ) (cid:1) + ( x − ⌊ x ⌋ − ) sinh( λ/
2) cosh (cid:0) λ ( x − ⌊ x ⌋ − ) (cid:1) sinh( λ/ . If µ is a signed Borel measure satisfying (H1’) - (H2) we define the odd 1-periodic function e F µ ( x ) := Z ∞ e h ( λ, x ) µ ( λ ) d λ. (4.13)Note that we do not have to assume (H4) in order to define e F µ in (4.13) since, for all x ∈ R , the function λ e h ( λ, x ) is O ( λ ) as λ →
0. If, however, we have (H4), the function F µ is well-defined and we have e F µ ( x ) = F µ ( x ) − F µ ( − x ) = X n ∈ Z e f µ ( x + n ) , verifying that e F µ is in fact the periodization of e f µ . We note that e F µ is differentiable for x / ∈ Z . For 0 ≤ x ≤ e h ( λ, x ) = h ( λ, x ) − h ( λ, − x )= xe − λx − (1 − x ) e − λ (1 − x ) + e − λ (cid:0) − e − λ (cid:1) (cid:16) xe − λx (cid:0) − e − λ (cid:1) + e − λx − (1 − x ) e − λ (1 − x ) (cid:0) − e − λ (cid:1) − e − λ (1 − x ) (cid:17) , (4.14)and we may use dominated convergence in (4.13) together with the computation in (1.11) to conclude that,under (H1’) - (H2) - (H3), we have e F µ (0 ± ) = ± . We highlight the fact that when µ is the Dirac delta measure, we recover the sawtooth function (multipliedby −
2) in (4.13). In fact, observing that for x / ∈ Z we have e h ( λ, x ) = − ∂∂λ sinh (cid:0) − λ ( x − ⌊ x ⌋ − ) (cid:1) sinh( λ/ ! , e find, for x / ∈ Z , e F µ ( x ) = Z ∞ e h ( λ, x ) d λ = − x − ⌊ x ⌋ − ) . This is expected since the corresponding e f µ is the signum function. In particular, the results we presentbelow extend the work of Li and Vaaler [26] on the sawtooth function.4.2.2. Main results.
The following two results provide a complete solution of the extremal problem (1.21) -(1.22) for the periodic functions F µ and e F µ defined in (4.11) and (4.13), with respect to arbitrary nontrivialprobability measures ϑ . This completes the framework initiated in [9], where this extremal problem wassolved for an analogous class of even periodic functions with exponential subordination. In what follows welet ϕ N +1 ( z ) = ϕ N +1 ( z ; d ϑ ) be the ( N + 1)-th orthonormal polynomial in the unit circle with respect to thismeasure and consider K N , A N +1 , B N +1 as defined in (4.9) and (4.10). Theorem 12.
Let µ be a signed Borel measure on R satisfying (H1’) - (H2) - (H4) , and let F µ be definedby (4.11) . Let ϑ be a nontrivial probability measure on R / Z and N ∈ Z + . (i) If L : C → C is a real trigonometric polynomial of degree at most N such that L ( x ) ≤ F µ ( x ) (4.15) for all x ∈ R / Z , then Z R / Z L ( x ) d ϑ ( x ) ≤ F µ (0) K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . (4.16) Moreover, there is a unique real trigonometric polynomial L µ : C → C of degree at most N satisfying (4.15) for which the equality in (4.16) holds. (ii) Assume that µ also satisfies (H3) . If M : C → C is a real trigonometric polynomial of degree at most N such that F µ ( x ) ≤ M ( x ) (4.17) for all x ∈ R / Z , then Z R / Z M ( x ) d ϑ ( x ) ≥ F µ (0 + ) K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . (4.18) Moreover, there is a unique real trigonometric polynomial M µ : C → C of degree at most N satisfying (4.17) for which the equality in (4.18) holds. Theorem 13.
Let µ be a signed Borel measure on R satisfying (H1’) - (H2) - (H3) , and let e F µ be definedby (4.13) . Let ϑ be a nontrivial probability measure on R / Z and N ∈ Z + . (i) If L : C → C is a real trigonometric polynomial of degree at most N such that L ( x ) ≤ e F µ ( x ) (4.19) for all x ∈ R / Z , then Z R / Z L ( x ) d ϑ ( x ) ≤ − K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 e F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . (4.20) oreover, there is a unique real trigonometric polynomial e L µ : C → C of degree at most N satisfying (4.19) for which the equality in (4.20) holds. (ii) If M : C → C is a real trigonometric polynomial of degree at most N such that F µ ( x ) ≤ M ( x ) (4.21) for all x ∈ R / Z , then Z R / Z M ( x ) d ϑ ( x ) ≥ K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 e F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . (4.22) Moreover, there is a unique real trigonometric polynomial e M µ : C → C of degree at most N satisfying (4.21) for which the equality in (4.22) holds. Periodic interpolation.
Before we proceed to the proofs of Theorems 12 and 13 we state and provethe periodic version of Proposition 6. Below we keep the notation already used in Section 2.
Proposition 14.
Let F be a -periodic Laguerre-P´olya function of exponential type τ ( F ) . Assume that F has a double zero at the origin and that F ( α F / > . Let µ be a signed Borel measure on R satisfying (H1’)- (H2) - (H4) , and let F µ be defined by (4.11) . (i) The functions x L ( F, µ, x ) and x M ( F, µ, x ) belong to L ( R ) . (ii) Define the trigonometric polynomials L ( F, µ, z ) = X | k | < τ ( F )2 π b L ( F, µ, k ) e ( kz ) (4.23) and M ( F, µ, z ) = X | k | < τ ( F )2 π c M ( F, µ, k ) e ( kz ) . (4.24) Then we have F ( x ) L ( F, µ, x ) ≤ F ( x ) F µ ( x ) ≤ F ( x ) M ( F, µ, x ) (4.25) for all x ∈ R . (iii) Moreover, L ( F, µ, ξ ) = F µ ( ξ ) = M ( F, µ, ξ ) (4.26) for all ξ ∈ R \ Z with F ( ξ ) = 0 . At ξ ∈ Z we have L ( F, µ, ξ ) = F µ (0) and M ( F, µ, ξ ) = F µ (0) + 1 . (4.27) Proof.
We have already noted, from (2.10), that z L ( F, µ, z ) and z M ( F, µ, z ) are entire functions ofexponential type at most τ ( F ). From (2.22) we find that | L ( F, µ, x ) | + | M ( F, µ, x ) | ≪ f µ ( x ) + 1 + | F ( x ) | x for x ∈ R . Since F is 1-periodic, it is bounded on the real line. Hence, in order to prove (i), it suffices toverify that f µ ∈ L ( R ). This is a simple application of Fubini’s theorem and conditions (H1’) - (H2) - (H4).In fact, Z ∞ f µ ( x ) d x = Z ∞ Z ∞ xe − λx µ ( λ ) d λ d x = Z ∞ λ µ ( λ ) d λ < ∞ . his establishes (i). The Paley-Wiener theorem implies that the Fourier transforms b L ( F, µ, t ) = Z ∞−∞ L ( F, µ, x ) e ( − tx ) d x and c M ( F, µ, t ) = Z ∞−∞ M ( F, µ, x ) e ( − tx ) d x are continuous functions supported in the compact interval [ − τ ( F )2 π , τ ( F )2 π ]. By a classical result of Planchereland P´olya [33], the functions z L ′ ( F, µ, z ) and z M ′ ( F, µ, z ) also have exponential type at most τ ( F )and belong to L ( R ). Therefore, the Poisson summation formula holds as a pointwise identity and we have L ( F, µ, x ) = X | k | < τ ( F )2 π b L ( F, µ, k ) e ( kx ) = X n ∈ Z L ( F, µ, x + n ) (4.28)and M ( F, µ, x ) = X | k | < τ ( F )2 π c M ( F, µ, k ) e ( kx ) = X n ∈ Z M ( F, µ, x + n ) . (4.29)Using the fact that F µ ( x ) = X n ∈ Z f µ ( x + n )for all x ∈ R , (4.25), (4.26) and (4.27) now follow from (4.28), (4.29) and Proposition 6, since F is 1-periodic.This establishes (ii) and (iii). (cid:3) Proof of Theorem 12.
Recall that we have normalized our orthonormal polynomials ϕ N +1 in orderto have ϕ N +1 (1) ∈ R . This implies that B N +1 (1) = 0.4.4.1. Optimality. If L : C → C is a real trigonometric polynomial of degree at most N such that L ( x ) ≤ F µ ( x )for all x ∈ R / Z , from Proposition 11 we find that Z R / Z L ( x ) d ϑ ( x ) = X ξ ∈ R / Z B N +1 ( e ( ξ ))=0 L ( ξ ) K N ( e ( ξ ) , e ( ξ )) ≤ F µ (0) K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . (4.30)This establishes (4.16). Under (H3) recall that we have F µ (0 + ) = F µ (0 − ) + 1 = F µ (0) + 1 . In an analogous way, using Proposition 11, it follows that if M : C → C is a real trigonometric polynomialof degree at most N such that F µ ( x ) ≤ M ( x )for all x ∈ R / Z then Z R / Z M ( x ) d ϑ ( x ) = X ξ ∈ R / Z B N +1 ( e ( ξ ))=0 M ( ξ ) K N ( e ( ξ ) , e ( ξ )) ≥ F µ (0 + ) K N (1 ,
1) + X ξ ∈ R / Z ; ξ =0 B N +1 ( e ( ξ ))=0 F µ ( ξ ) K N ( e ( ξ ) , e ( ξ )) . This establishes (4.18).4.4.2.
Existence.
Define the trigonometric polynomial B N +1 ( z ) = B N +1 ( e ( z )) B N +1 ( e ( z )) . (4.31) ince the polynomial B N +1 has degree N + 1 and has only simple zeros in the unit circle, we conclude thatthe trigonometric polynomial B N +1 has degree N + 1, is nonnegative on R and has only double real zeros.Since every trigonometric polynomial is of bounded type in the upper half-plane U , it follows by [1, Problem34] that B N +1 is a Laguerre-P´olya function.We now use Proposition 14 to construct the functions L µ ( z ) := L ( B N +1 , µ, z ); M µ ( z ) := M ( B N +1 , µ, z ) . Since B N +1 has exponential type 2 π ( N + 1) we see from (4.23) and (4.24) that L µ and M µ are trigonometricpolynomials of degree at most N . Since B N +1 is nonnegative on R we conclude from (4.25) that L µ ( x ) ≤ F µ ( x ) ≤ M µ ( x )for all x ∈ R / Z . Moreover, from (4.26), (4.27) and the quadrature formula given by Proposition 11, weconclude that the equality in (4.16) holds. Under the additional condition (H3), we use (4.12) to see thatthe equality in (4.18) also holds.4.4.3. Uniqueness. If L : C → C is a real trigonometric polynomial of degree at most N satisfying (4.15) forwhich the equality in (4.16) holds, from (4.30) we must have L ( ξ ) = F µ ( ξ ) = L µ ( ξ )for all ξ ∈ R / Z such that B N +1 ( e ( ξ )) = 0. Since F µ is differentiable at R / Z − { } , from (4.15) we must alsohave L ′ ( ξ ) = F ′ µ ( ξ ) = L ′ µ ( ξ )for all ξ ∈ R / Z − { } such that B N +1 ( e ( ξ )) = 0. These 2 N + 1 conditions completely determine a trigono-metric polynomial of degree at most N , hence L = L µ . The proof for the majorant is analogous.4.5. Proof of Theorem 13.
Optimality and uniqueness.
These follow exactly as in the proof of Theorem 12 using the fact that e F µ (0 ± ) = ± . Existence.
We proceed with the construction of the extremal trigonometric polynomials in two steps:
Step 1 . Suppose that µ satisfies (H4).In this case we know that e F µ ( x ) = F µ ( x ) − F µ ( − x ) (4.32)for all x ∈ R . With the notation of Proposition 14 and B N +1 given by (4.31), we define e L µ ( z ) = L ( B N +1 ( z ) , µ, z ) − M ( B N +1 ( − z ) , µ, − z ) (4.33)and e M µ ( z ) = M ( B N +1 ( z ) , µ, z ) − L ( B N +1 ( − z ) , µ, − z ) . (4.34)It is clear from (4.32) and (4.25) that e L µ ( x ) ≤ e F µ ( x ) ≤ e M µ ( x ) (4.35) or all x ∈ R / Z . Moreover, from (4.26) and (4.27) we find that e L µ ( ξ ) = e F µ ( ξ ) = e M µ ( ξ ) (4.36)for all ξ ∈ R / Z − { } such that B N +1 ( e ( ξ )) = 0 and e L µ (0) = − e M µ (0) = 1 . (4.37)Using the quadrature formula given by Proposition 11, we see that equality holds in (4.20) and (4.22). Step 2 . The case of general µ .For every n ∈ N we define a measure µ n given by µ n (Ω) := µ (cid:0) Ω − n (cid:1) , where Ω ⊂ R is a Borel set. Note that µ n satisfies (H1’) - (H2) - (H3) - (H4). Let e F n := e F µ n , and e L n := e L µ n and e M n := e M µ n as in (4.33) and (4.34). Since properties (4.35), (4.36) and (4.37) hold for each n ∈ N ,in order to conclude, it suffices to prove that e F n converges pointwise to e F µ and that e L n and e M n convergepointwise (passing to a subsequence, if necessary) to trigonometric polynomials e L µ and e M µ .Observe first that e F n ( x ) = Z ∞ e h (cid:0) λ + n , x (cid:1) µ ( λ ) d λ (4.38)for all x ∈ R . From (4.14) we see that, for 0 ≤ x ≤ (cid:12)(cid:12)(cid:12)e h (cid:0) λ + n , x (cid:1)(cid:12)(cid:12)(cid:12) ≤ xe − λx + (1 − x ) e − λ (1 − x ) + r ( λ ) , (4.39)where r ( λ ) is O (1) for λ < O ( e − λ ) for λ ≥
1, uniformly in x ∈ [0 ,
1] and n ∈ N . For any x ∈ [0 , L ( R + , µ ( λ ) d λ ), and therefore we may use dominated convergencein (4.38) to conclude that e F n ( x ) → e F µ ( x ) as n → ∞ .From (2.22), (4.28) and (4.29) we find that e M n ( x ) − e L n ( x ) = 4 B N +1 ( x ) B ′′ N +1 (0) X k ∈ Z x + k ) . (4.40)for all x ∈ R . Note that the right-hand side of (4.40) is bounded since B N +1 is a trigonometric polynomialwith a double zero at the integers. Therefore, we arrive at − B N +1 ( x ) B ′′ N +1 (0) X k ∈ Z x + k ) + e F n ( x ) ≤ e L n ( x ) ≤ e F n ( x ) ≤ e M n ( x ) ≤ e F n ( x ) + 4 B N +1 ( x ) B ′′ N +1 (0) X k ∈ Z x + k ) . From (4.38) and (4.39) we see that (cid:12)(cid:12)e F n ( x ) (cid:12)(cid:12) ≤ Z ∞ xe − λx µ ( λ ) d λ + Z ∞ (1 − x ) e − λ (1 − x ) µ ( λ ) d λ + Z ∞ r ( λ ) µ ( λ ) d λ ≤ C for all x ∈ [0 ,
1] and n ∈ N , since each of the first two integrals is a continuous function of x ∈ (0 , x → x →
1, due to condition (H3) and the computation in (1.11). This implies that e L n and e M n are uniformly bounded on R . The 2 N + 1 Fourier coefficients of e L n and e M n are then uniformlybounded on R and we can extract a subsequence { n k } such that e L n k → e L µ and e M n k → e M µ uniformly incompact sets, where e L µ and e M µ are trigonometric polynomials of degree at most N . This completes theproof. cknowledgements E. C. acknowledges support from CNPq - Brazil grants 302809 / − / −
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