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Recursiveness, Positivity, and Truncated Moment Problems

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Recursiveness, Positivity, and Truncated Moment Problems HOUSTON JOURNAL OF MATHEMATICS Volume 17, No. 4, 1991 RECURSIVENESS, POSITIVITY, AND TRUNCATED MOMENT PROBLEMS I•A•L E. CURTO LAWRENCE A. FIALKOW Abstract. Using elementary techniques from linear algebra, we de- scribe a recursire model for singular positive Hankel matrices. We then use this model to obtain necessary and sufficient conditions for existence or uniqueness of positive Borel measures which solve the truncated moment problems of Hamburger, Hausdorff and Stieltjes. We also present analogous results concerning Toeplitz matrices and the truncated trigonometric moment problem. Dedicated to the memory of Domingo A. Herrero 1. Introduction. Given an infinite sequenceof complexnumbers 7 - {70,71,--' } and a subsetK C_C, the K-powermoment problem with data 7 entails finding a positiveBorel measurey on C suchthat (1.1) / tJdp(t)=7j (J •- O) and supp p C_K. The classicaltheorems of Stieltjes,Toeplitz, Hamburger and Hausdorffpro- vide necessaxyand sufficientconditions for the solubilityof (1.1) in case K = [0,+o•),K ---T := (t e C: Itl- 1},K - • andK - [a,bl(a,b e [•), respectively([AhKI, [Akhl, [KrNI, [Lanl, [Sat], [ShT]). For example, when K = •, Hamburger'sTheorem implies that (1.1) is solubleif and Both authorswere partially supportedby grantsfrom NSF. Rafil Curto was alsopartially supportedby a Universityof Iowa faculty scholaraward. 603 604 R. E. CURTO AND L. A. FIALKOW onlyif the Hankelmatrices A(n):= (7i+j)in,j=o(n = 0,1,... ) arepositive semidefinite[ShT, Theorem1.1.2]. The classicaltheory also providesuniqueness theorems and parame- terizationsof the setsof solutions.Parts of the theory havebeen extendedto covermore general support sets for/• ([SEMI, [Gas],[Sch]), and the "mul- tidimensionalmoment problem" has also been studied extensively (lAtz], [Ber], [Fug],[Put]). For 0 _• m < •x•,let ? = (7o,..-,7m) E Cre+l, and considerthe truncated K-power moment problem (1.2) / tJdl•(t)=7j (0 _< j <_ m). and supp/• C_K. For example, the "even" caseof the truncated Hamburgerproblem corre- spondsto m = 2k for somek _>0 and K = It. On the basisof Hamburger's Theorem,one might surmise that the criterionfor solubilityof (1.2) in this caseis A(k) _>O. As wewill seein the sequel,this positivity condition is al- waysnecessary, and is sufficientwhen A(k) is nonsingular[ABE, Theorem 1.3];however, when A(k) is singular,the positivity of A(k) is not sufficient. In particular, the classicaltheory for the "full" moment problem does not include the theory of the truncated momentproblem as a specialcase. On the other hand, it is knownthat the theoryof the truncatedmoment prob- lem can be used to solve the full moment.problem and to recover such resultsas Hamburger'sTheorem [Lan, p. 5], [Akh]. Observethat the truncatedStieltjes moment problem differs slightly fromthe classicaltruncated Stieltjes moment problem for 7 = (70,..., 7m), defined by: øøtJdl•(t)= 7j (O_< j _<m-1) and (1.3) tmd[2(t)_• •m. For purposesof parameterizingthe solutionspaces, the classicalproblem has provedmore amenablethan the problemwe consider.However, for cer- tain basicinterpolation problems in operatortheory (discussed below), it is TRUNCATED MOMENT PROBLEMS 605 necessaryto considerthe Stieltjesmoment problem of (1.2), i.e., wherethe inequalityin (1.3) is replacedby equality.Despite the importanceof trun- cated moment problems,their treatment in the literature seemssomewhat fragmentary. Most sourcesignore the "odd" caseentirely, although for us this caseis fundamental.The usualapproach (e.g. [Ioh, TheoremA.II.1], [ShT, p. 28 if.I) is to treat the "nonsingulareven" case of Hamburger's problem and then reducethe "singulareven" caseto the nonsingularcase by means of quasi-orthogonalpolynomials and Gaussianquadrature. In the present note we present a comprehensiveand unified treat- ment of necessaryand sufficientconditions for the solubilityof (1.2) in case K - R, [a,b], [0,-t-•x•) or T. Our approachrevolves around the notionof "recursiveness"for positiveHankel and Toeplitz matrices(Theorems 2.4 and 6.4). This key ingredient,apparently neglected in the literature,allows us to recover many classicalresults and to obtain many new ones. Our basic result, Theorem 3.1, providessimple operator-theoreticnecessary and sufficientconditions for solubility of the "odd" caseof the truncated Ham- burger moment problem. Our solution is based on elementary techniques concerningnonnegative polynomials, Lagrange interpolation, and linear al- gebra (Proposition3.3); for the "singualrodd" case,the solutionrests on a descriptionof positive singularHankel matrices. Theorem 2.4 showsthat eachentry of sucha matrix (exceptperhaps the lowerright-hand entry!) is recursivelydetermined by the entriesin the largestnonsingular upper left-hand"corner" of the matrix. Theorem2.4 immediatelyallows us to re- ducesingular Hamburger moment problems to equivalentnonsingular ones without recourseto quasi-orthogonalpolynomials or Gaussianquadrature (Corollary3.4). The "even"cases of Hamburger'sproblem are then treated as consequencesof the "odd"cases (Theorem 3.9). Theorems3.8 and 3.10 contain our uniquenesscriteria for the truncatedHamburger problem. So- lutions of the Hausdorffand Stieltjesmoment problems are corollariesof this approachand are presentedin Sections4 and 5, respectively.In Sec- tion 6 we treat the trigonometricmoment problem (K - T) througha corresponding(but independent)analysis of positiveTocplitz matrices. The presentwork supplements[CuF], where,as part of a detailed study of k-hyponormalityand quadratic hyponormalityfor unilateral weightedshifts, we solvedthe following"subnormal completion problem" of J. G. Stampfii[Sta]: When may a finite sequenceof positivenumbers, 70,... ,7m, be "completed"to a sequence? t%)n=0 whichis the mo- 606 R. E. CURTO AND L. A. FIALKOW ment sequenceof a subnormalunilateral weighted shift Wv ? In [CF] the subnormalityof Wv was establishedusing Berger's Theorem [Con]' We essentiallysolved the truncated Stieltjes moment problem with a measure producedfrom the spectral measureof a normal operatorßOur motivation for the present work was the desireto developa more transparent method of solving such truncated moment problems,without recourseto such an- alytic tools as the spectral theorem or quasi-orthogonalpolynomials: our new treatment satisfiesthis requirement. In particular, Section 3 concludes with a simplecomputational algorithm for solvingthe truncated Hamburger problem. 2. Recursire Models for Positive Hankel Matrices. Our goal in this section is to prove a structure theorem for positive semi-definiteHankel matrices.If A is a Hankelmatrix of sizek+ 1, let 7 = (70,..., 72k)E R2k+• representthe entriesof A, in the sensethat A = Av := (Ti+5)i,j=O'k The j-th columnof A will be denotedbyvj := (7j+e)e=o,k 0 _<j _<k, sothat A canbe briefly written as (v0 ..-v&). More generally, letv(i,j) := (7i+e)i=o; observethat vj = v(j, k). Sincewe'll be mainly interestedin positive semi-definite Hankel matrices, we shall always assumethat T0 > 0. The (Hankel) rank of 7, denotedrank 7, is now definedas follows: If A is nonsingular,rank 7 := k + 1; if A is singular,rank 7 is the smallestinteger i, 1 _<i < k, suchthat vi E span(v0,... ,vi_•). Thus, if A is singular, i.e., rank7 -< k, thereexists a unique(I> := (I>(7)= (9'o,..., 9'i-•) • Ri suchthat vi = ½*ovo+ ." + •i-lVi-1 ß When A >_0 (i.e., A is positive semidefinite),rank 7 admits a usefulalternate description, which depends on the following basic facts about Hankel matrices. Lemma 2.1. LetT=(To,...,72•) • R2•+1,70> 0, andIetA=A v. For 0 <_m _<k, let A(m):= (Ti+j)i,r•=o(A(m) is theupper/e/t-hand corner oœ A oœsize m + 1, and A = A(k)). (i) (cf. /CF, Proposition2.7]) If A is positiveand invertible, so is A(j) œorevery j _<k. (ii) (cf. /CF, Proposition2.12]) Let r := rank7. Then A(r- 1) is invertible, and if r _< k, we have 9,(•/)= A(r - 1)-•v(r, r - 1). Proof: (ii) The fact that A(r- 1) is invertibleis containedin [CF, Propo- sition 2.12]ß To provethe identityfor (I'(7), let (I,' •-- (9,•,..., •v--1)' :-- TRUNCATED MOMENT PROBLEMS 607 A(r- 1)-Xv(r,r - 1). We employthe minimality condition in the definition of r - rank V'•(V)- (T0,...,Tr-x) E Rr is the uniquesolution of In particular,•0?j + '" + •?-x?j+?-x = ?j+?,0 _<j _<r - 1; also,from the definitionof •, •?j + -.. + •_•?j+•_• = ?j+•,0 < j < r - 1, so the invertibilityof A(r- 1) impliesthat •'= •(?). [] We can nowrelate rank • to a simplecriterion concerning singularity. Proposition 2.2. Let A : (Ti+j)i,j=Obe a positivesemi-definite Hankel matrix. Assumethat A is singular. Then rank ? = m/n {j ß 1 _<j _< k and A(j) is singular}. Proof: Let r '= rank?. By Lemma2.1 (ii), A(r- 1) is invertible.Since A _>0, Lemma2.1 (i) impliesthat A(j) is invertiblefor 0 _<j _<r - 1. On the otherhand, the definitionof r impliesthat A(r) is singular(the columns are dependent),so r = min {j- 1 _<j <_k andA(j) is singular}. [] We proceednow to describea model for positiveHankel matrices, which revealsa degreeof recursivenesspresent in any singularpositive Hankelmatrix. For ? = (?0,.--,?2k) let v(k + 1,k) := (')'k+l+j)j=k O. For •'2k+2 • • and5 = (,0,. ßß ,')'2k,,2k+l), let • := A• denotethe Hankel extension ofA givenby • -- f["{r+SJr,s=O •u+x= (v(k -•- A1, k)* v(k+"{2k+21,k)) ' The followingelementary, but veryuseful, result will be neededin the proof of Theorem2.4; an operatorversion of this resultis implicit in the work of Smul'jan[Smu] (cf. [CuF]). Lemma 2.3. (cf. /CuF, Proposition2.3]) Let A • Mn(C),b • Cn, c • C, and let •:= b* c ' (i) If • > O,then A _>0, b = Aw forsome w • C'•, and c > w*Aw. 608 R. E. CURTO AND L. A. FIALKOW (ii) LetA> -- Oandb=Awforsomew6C n. Then fi>__ 0if and only if c _>w* Aw. (iii) IœA >_0 andb = Aw, thenrank fi = rankA if andonly if c = w*Aw. We now presentour structure theorem for positive Hankel matrices.
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