In 2002 C. Berg, Y. Chen, and M. Ismail found a nice relation between the
determinancy of the Hamburger moment problem and asymptotic behavior of the
smallest eigenvalues of the corresponding Hankel matrices. We investigate
whether an analog of this statement holds for the Nevanlinna--Pick
interpolation problem.Comment: 5 pages, LaTeX fil

Golinskii, L.

Peherstorfer, F.

Yuditskiy, P.

Zurnal matematiceskoj fiziki analiza geometrii

English

arXiv

In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinacy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna-Pick interpolation problem

Yuditskii, P.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Journal of Mathematical Physics, Analysis, Geometry2008, vol. 4, No. 4, pp. 451456On the BergChenIsmail Theoremand the NevanlinnaPick ProblemL. GolinskiiMathematical Division, B. Verkin Institute for Low Temperature Physics and EngineeringNational Academy of Sciences of Ukraine47 Lenin Ave., Kharkiv, 61103, UkraineE-mail:leonid.golinskii@gmail.comF. Peherstorfer and P. YuditskiiInstitute for Analysis, Johannes Kepler University LinzA-4040 Linz, AustriaE-mail:franz.peherstorfer@jku.atpetro.yuditskiy@jku.atReceived May 19, 2008In 2002, C. Berg, Y. Chen, and M. Ismail found a nice relation betweenthe determinacy of the Hamburger moment problem and asymptotic beha-vior of the smallest eigenvalues of the corresponding Hankel matrices. We in-vestigate whether an analog of this statement holds for the NevanlinnaPickinterpolation problem.Key words: moment problem, Blaschke product, Carleson measure, Pickmatrix.Mathematics Subject Classication 2000: 30E05 (primary), 30D50(secondary).The Hamburger moment problem is a problem of nding conditions ona sequence fsjg, j = 0; 1; : : : ; so that there exists a positive Borel measure  andsj=ZRxjd(x); j = 0; 1; : : : : (1)With  one can associate the innite Hankel matrix and the sequence of itsprincipal submatricesH = ksi+jk1i;j=0; Hn= ksi+jkni;j=0; n = 0; 1; 2; : : : : (2)Partially supported by the Austrian Founds FWF (Project P20413N18) and Marie CurieInternational Fellowship within the 6th European Community Framework Programme (ContractMIF1-CT-2005-00696).c L. Golinskii, F. Peherstorfer, and P. Yuditskii, 2008L. Golinskii, F. Peherstorfer, and P. YuditskiiBy the famous result of Hamburger (1) has a solution if and only if Hn 0 forall n.Denote by fn;jgnj=0the eigenvalues of Hn(2) labelled in increasing order.Because of the interlacing property we have0  n+1;0 n;0 n+1;1 n;1 : : :  n+1;n n;n n+1;n+1; (3)and so for each k n;kare monotone decreasing, k(H) = limn!1n;kexists, and0  0(H)  1(H)  : : : : (4)Let us call the problem (1) regular if 0(H) > 0, and singular otherwise.In 2002, C. Berg, Y. Chen, and M. Ismail [2] proved a beautiful result which statesthat (1) is regular if and only if it has innitely many solutions (indeterminate).The Hamburger moment problem is one of the representatives of the so-calledclassical interpolation problems [1]. In this note we address to another one, speci-cally, the NevanlinnaPick interpolation problem in the Schur class S of functionscontractive and analytic in the unit disk D . This is a problem of nding thesolutions off(zk) = wk; k = 0; 1; 2; : : : ; (5)where zkare distinct points in D , wkcomplex numbers, and f 2 S. The well-known criterion for (5), to have at least one solution, is given in terms of Pickmatrices byPn:=1  wiwj1  zizjni;j=0 0for all n = 0; 1; : : : . For the eigenvalues fn;jgnj=0of the matrices Pnlabelledin increasing order the above relations (3), (4) hold, and, again, we distinguishbetween the regular and singular NevanlinnaPick problem, i.e., in this paper wesay that the problem (5) is regular if 0(P ) > 0, and it is singular if 0(P ) = 0.With respect to a number of various questions there is a strong similaritybetween dierent classical interpolation problems, that is, if one can prove thisor that statement with respect to one of the classical problems, a quite parallelstatement holds for another one. In this note we study the question: Is it true thata NevanlinnaPick problem has innitely many solutions (indeterminate) if andonly if 0(P ) > 0? We give a negative answer to this question. More precisely,we construct data fzk; wkg1k=0of an indeterminate NevanlinnaPick problem suchthat 0(P ) = 0.First of all, we note that the Blaschke condition on the interpolation nodesZ = fzkg1Xk=0(1  jzkj) <1 (6)452 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4On the BergChenIsmail Theorem and the NevanlinnaPick Problemguarantees that the interpolation problemf(zk) = 0; k = 0; 1; 2; : : : ; (7)has innitely many solutions. Indeed, every function of the formf = g(z)B(z);where g(z) 2 S, and B(z) is the Blaschke productB(z) =Ykzk  z1  zzkjzkjzk;solves (7). Note that (6) is necessary and sucient for the problem (7) to beindeterminate.Thus, our goal is to construct a Blaschke set Z such that 0(P ) = 0 for thesequence of NevanlinnaPick matrices of the specic form Pn= Kn, whereKn:=11  zizjni;j=0:In fact, our main statement here characterizes completely the regularity of suchNevanlinnaPick problems in the above sense.Recall (see, e.g., [4]), that a (nite) Borel measure  on D is called a Carlesonmeasure, ifZDjf j2d  CZTjf j2dm (8)for all f 2 H2. Here dm is the normalized Lebesgue measure on the unit circle T.Due to Carleson's theorem [3] such measures are characterized completely by thefollowing property: there exists C > 0 such that(Q())  Cfor all   <   , 0 <  < 1,Q() := fz 2 D : j arg z   j  ; 1    jzj < 1g:Theorem 1. Let Z satisfy (6), B(z) be the corresponding Blaschke product.The NevanlinnaPick problem (7) is regular if and only if the measure , dened by(fzkg) = jB0(zk)j 2; (9)is a Carleson measure in D .Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 453L. Golinskii, F. Peherstorfer, and P. YuditskiiP r o o f. Let  (9) be a Carleson measure. For arbitrary c0; : : : ; cn2 C puth(z) = B(z)nXk=0ckz   zk2 H2withkhk2=nXi;j=0Kijcicj; K = kKijk1i;j=0=11  zizj1i;j=0:We have h(zj) = cjB0(zj) for j = 0; 1; : : : ; n and h(zj) = 0 for j  n+ 1, soZDjhj2d =nXj=0jcjj2:Hence by (8)ZTjhj2dm = khk21CnXj=0jcjj2;and so 0(K)  C 1> 0, as claimed.Conversely, assume that the NevanlinnaPick problem in question is regular.We use the standard orthogonal decomposition L2(T) = H2LH2 . PutKB= (BH2)?= H2\BH2 ; k(z) =B(z)z   zk; k = 0; 1; : : : :It is easy to see that fkg is complete in KB. Indeed, (z   zk) 12 H2 , sok2 KB. Let g 2 KB, g?kfor all k = 0; 1; : : : . Then g = Bg1(), g12 H2,and0 = (g; k) =ZTB()   zkB()g1() dm = g1(zk);that is, g12 BH2, so g 2 H2 and g  0, as claimed. Hence the system offunctionsf(z) = B(z)nXk=0ckz   zk+B(z)g(z) = h(z) +B(z)g(z); n = 0; 1; : : : ;c0; : : : ; cn2 C , and g 2 H2, is dense in H2. We prove (8) for such functions.As above,f(zj) = cjB0(zj); j = 0; 1; : : : ; n; f(zj) = 0; j  n+ 1;454 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4On the BergChenIsmail Theorem and the NevanlinnaPick ProblemandZDjf j2d =nXj=0jcjj2; kfk2= khk2+ kgk2 khk2=nXi;j=0Kijcicj;and soZTjf j2dm nXi;j=0Kijcicj 0(K)nXj=0jcjj2= 0(K)ZDjf j2d:The proof is complete.If the NevanlinnaPick problem (7) is singular (as in the example below), thenso is the general problem (5). Indeed, for D = diag(w0; w1; : : : ; wn)Pn= Kn DKnD Kn;and so 0(P )  0(K).E x a m p l e. Put zk= 1 k p, p > 1. Evidently Z is a Blaschke set. On theother hand(1  jznj2)jB0(zn)j =n 1Yk=1zn  zk1  znzk1Yk=n+1zk  zn1  znzk1Yk=n+1kp  npnp+ kp  11Yk=n+11 nkp exp  1Xk=n+1nkp! exp n+ 12p(p  1);sojB0(zn)j 21n2pexpn+ 12p 1(p  1):Thus the measure  (9) is innite and, moreover, it is not of Carleson type.There is a simple way of manufacturing regular NevanlinnaPick problems(7). Recall that Z = fzkg is the Carleson (uniformly separated) sequence ifÆ(Z) := infnYk 6=njzkjzkzn  zk1  znzk> 0: (10)Assume that Z satises (10). By the theorem of H.S. Shapiro and A.L. Shields[5] the system of functionsxk(z) =(1  jzkj2)1=21  zkzk = 0; 1; : : : ;Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4 455L. Golinskii, F. Peherstorfer, and P. Yuditskiiforms a Riesz basis in KB. So, for all c0; : : : ; cn2 C there is c > 0 such thatnXi;j=0Kij(1  jzij2)1=2(1  jzjj2)1=2cicj cnXj=0jcjj2;ornXi;j=0Kijdidj cnXj=0jdjj2(1  jzjj2) cnXj=0jdjj2;as claimed.Acknowledgement. The authors thank Alexander Kheifetz for helpful dis-cussions. The paper is written mainly during the rst author's visit to JohannesKepler University, Linz. He wishes to thank JKU for the hospitality and theMarie Curie Foundation that made this visit possible.References[1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions inAnalysis. Oliver and Boyed, Edinburgh, 1965.[2] C. Berg, Y. Chen, and M. Ismail, Small Eigenvalues of Large Hankel Matrices:the Indeterminate Case.  Math. Scand. 91 (2002), No. 1, 6781.[3] L. Carleson, An Interpolation Problem for Bounded Analytic Functions.  Amer.J. Math. 80 (1958), No. 4, 921930.[4] J.B. Garnett, Bounded Analytic Functions. (Revised rst edition.) Graduate Textsin Mathematics. 236. Springer, New York, 2007. xiv+459 pp.[5] H.S. Shapiro and A.L. Shields, On Some Interpolation Problems for AnalyticFunctions.  Amer. J. Math. 83 (1961), No. 3, 513532.456 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 4

On the Berg--Chen--Ismail theorem and the Nevanlinna-Pick problem

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