an interpolation problem for coefficients of h∞ functions

8/2/2019 An Interpolation Problem for Coefficients of H Functions

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An Interpolation Problem for Coefficients of H FunctionsAuthor(s): John J. F. FournierSource: Proceedings of the American Mathematical Society, Vol. 42, No. 2 (Feb., 1974), pp.402-408

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COEFFICIENTSOF HO1 FUNCTIONS 403This identity plays the same role here as the parallelogram law does in themethod of Shapiro and Rudin [10, p. 35] and [7, pp. 855-856].

THEOREM. Let {nk}k'O be a sequence of nonnegative integers so that,for some 6> 0, nk+1 >(I +6)nk f or al k. Let O IVkI2


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404 J. J. F. FOURNIER [FebruaryThese assertions clearly hold when k=O. Supposing that (2) and (3)hold for some index k, considei the formula

gk+l(Z) = gk(Z) + Vk+?Z k+lkh(Z) = gk(Z) + fk+1(Z), say.By (3), the support offk+1 is contained in the interval [nk+l-nk nk+1].Hence(4) Gk+1 Gk U [fk+1- nk, nk+41c [0, nk+1], as required.Similarly Hk+1c [-nk+1, 0]. Thus assertions (2) and (3) hold for all k. Itfollows from (2) that gk E H'; so (b) holds.Next, because nk+1>2nk, the intervals [0, nk] and [nk+1-nk,nk+1] aredisjoint. Therefore kkandfk+1 have disjoint supports and(5) ki+(n) = g (n) for all n1.Before taking up the case when *S< we mention two more facts aboutthe polynomials gk. First, it follows from (5) that the weak star limit gdiscussed above is unique and that gk iS the partial sum of order nk of theFourier series of g. Indeed, since the subsequence of partial sums fgk }'=kis uniformly bounded, it follows, without appeal to weak starcompactness,that the corresponding trigonometricseries must be the Fourier series of abounded function [12, vol. 1, p. 148].Second, it is clear from (4) that Gkc {no}U [n1-nD, nl]U**. u[nk-nk_1 nkj- Therefore, the support of k is contained in {n0}U[nl-no, nl]u..U t[nk-nk-1, nJU-9 - - .When 6< I, we must split {nk} into a finite number of subsequences, sothat the above method works for each subsequence, and so that thefunctions g corresponding to different subsequences have disjointlysupported transforms. The procedure is like the one for Riesz products[5, pp. 108-109] or [12, vol. II, p. 131].Let q be the smallest integer such that o(I + )" I>1. Split {n,} and{Vk} into q subsequences in each of which the index k runs through anarithmetic progression with period q. Let {nkJ} , be such a subsequence.Since ki71I-k =q, we have

1 1 + 2nkin > (1 + f nk (70?+ n>2k.


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19741 COEFFICIENTSF H' FUNCTIONS 405

by the choice of q. So the methodused aboveapplied o thissubsequenceyieldsa functiong in H' withg(nk )=Vki forall i. Moreover is supportedby {nklU [Ink, l-nk0 fnkIU*-)..) n[ki-nk, nlki-]U It is easy to verifythat nk-nki __>nki-1, so that the only term of the sequence {fnk}lyingin the interval nki-nk1 , nk1] is nk . Hence (nk)=O for all the termsof {nk} except henki. Thuswecan solvetheinterpolation roblem oreachof the q subsequences nd add the solutions to obtainan H' functionwhich solves the interpolationproblem or {nk} and {vk}. Thiscompletesthe proof.Wecan sharpen he conclusionof the Theorem n two standardways.First, the function g in H' can be chosen so that JJgJJ._(qe)112liV112where IIvII2=(2=ovkI2)1/2 and q is the integer definedabove. To seethis we begin with the basiccase when 6? I andq= l. There is nothingto prove if l1vI12=O;o let 11vI12>O.et g be the weak star limit point ofthe polynomialsk. By (a)

00

llgll0 - 171 (1 + IVkl2)1/2 < exp(IIVII2/2).k=OGiven {Vk}, define a new sequence {vk} by Vk=VkIIIV2.Applyingourmethodto thesequence vk} we obtaina functiong' in H' withg'(n )=vvfor all k and lIg'IK,


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406 J. J. F. FOURNIER [Februaryinformation about the interpolating function g as our method does,but which allow a weakening of the assumption on the sequence {fnk}.First, one can use a method of Paley [6, Lemma 2, pp. 124-126] to showthat if a set of a set of nonnegative integers has the LX interpolationproperty then it has the H' interpolation property. It is well known [8,p. 225] that a set of integers {nk} has the LX interpolation property if andonly if it is a A(2) set, that is if and only if every LI function whose trans-form is supported by {nk} is actually in L2. Thus, in order that a set ofnonnegative integers have the H' interpolation property it is necessaryand sufficient that this set be a A(2) set. In fact, S. A. Vinogradov hasshown that one can simultaneously solve the Rudin-Carleson interpolationproblem and the above interpolation problem [11].Unfortunately no arithmetic characterization of A(2) sets is known.It is known, however, that any union of finitely many Hadamard sets isa A(2) set and that some A(2) sets are not finite unions of Hadamard sets[8, p. 210]. Thus, in ?1, the hypothesis on {fnk} is stronger than necessary.But the proof given in ?1 is more elementary than those discussedabove.

The first proof that every Hadamard set has the LX interpolationproperty was indirect [1, Satz I, p. 212]. Subsequently, Salem andZygmund used Riesz products to give a simple construction of LXinterpolating functions [9]. We want to compare their construction withthe one given here. For simplicitysuppose that nk+l_3nk for all k; let00 I1kl2< cc. Consider the product00 0011 (1 + VkZflk - vZ-1k) I [1 + 2i Im(vkz k)].k=O k= 0

Expanding this product formally yields a trigonometric series which canbe shown to be the Fourierseriesof a function in LX with f (nk) =vkfor all k and lf12


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1974] COEFFICIENTSOF H* FUNCTIONS 407of a bounded function.In fact it is the Fourierseries of the functiongconstructed n ?1.It is not clearwhetherthis relationbetweeng and f can be used toprovethatg is bounded. Instead we have modifieda method due inde-pendently o Shapiro[10, p. 35] and Rudin [7]. The idea of setting uptwo sequencesof trigonometricpolynomialswith prescribed oefficientsis dueto them.What s newhere is theuseof the identity(1), rather hanthe parallelogramaw, to estimate the LX norms of the polynomials;this changeallows us to generatepolynomialswhosenonzerocoefficientsdo not all have the same absolutevalue.Themethodused here seemsto be dual to the Hilbert pacemethodof[3, Lemma2], but has the advantageof producingthe interpolatingfunction g explicitly. Otherwise he two methodsare equivalent n thesensethatanythingwhichcanbeprovedbyone of themethodscanalsobeproved by the other method. For instance, if nk+1>2nk for all k, and VkO0for all k, then the functiong constructedn ?1 has the property hat thesupportof g is exactly heset Gdefinedabove,and in any caseg is carriedby G; it was shownin [3, Remark8] thattheremust be an interpolatingfunctionwhose transform s carriedby G. Becauseof this equivalencemanyof the results n [3] can now be proved n two ways.In particularwe now have two proofs of Theorem1 of [3] but we still do not knowwhetherthe assumption hat I is an exponentialgap systemimpliesthat0= 1u(mk)12


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408 J. J. F. FOURNIER11. S. A. Vinogradov, TheBanach-Rudin-Carlesonnterpolation heoremand normsof embeddingoperators or certain classes of analyticfunctions, Zap. Naucn. Sem.Leningrad,Otdel. Mat. Inst. Steklov. (LOMI) 19 (1970), 6-54=Sem. Math. V. A.Steklov Math. Inst. Leningrad19 (1970), 1-28. MR 45 #4137.12. A. Zygmund, Trigonometric eries, 2nd rev. ed., Vols. I, II, CambridgeUniv.Press, Cambridge, 1959. MR 21 #6498.DEPARTMENT OF MATHEMATICS,UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER 8,

BRITISH COLUMBIA, CANADA

an interpolation problem for coefficients of h∞ functions

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