Mathematische Annalen (2021) 380:267–291
https://doi.org/10.1007/s00208-021-02166-x
Mathematische Annalen
The multidimensional truncated moment problem:
Carathéodory numbers from Hilbert functions
Philipp J. di Dio1·Mario Kummer2
Received: 18 May 2019 / Revised: 8 January 2021 / Accepted: 11 March 2021 / Published online: 24 March 2021
© The Author(s) 2021
Abstract
In this paper we improve the bounds for the Carathéodory number, especially on
algebraic varieties and with small gaps (not all monomials are present). We provide
explicit lower and upper bounds on algebraic varieties, Rn, and [0,1]n. We also treat
moment problems with small gaps. We find that for every ε>0 and d∈Nthere is
an∈Nsuch that we can construct a moment functional L:R[x1,··· ,xn]≤d→R
which needs at least (1−ε) ·n+d
natoms lxi. Consequences and results for the
Hankel matrix and flat extension are gained. We find that there are moment functionals
L:R[x1,··· ,xn]≤2d→Rwhich need to be extended to the worst case degree 4d,
˜
L:R[x1,··· ,xn]≤4d→R, in order to have a flat extension.
Mathematics Subject Classification 44A60 ·14P99 ·30E05 ·65D32 ·35R30
Contents
1 Introduction .............................................268
2 Preliminaries .............................................269
2.1 Truncated moment problem ...................................269
2.2 Algebraic geometry .......................................272
3 Carathéodory numbers for moment sequences with small gaps ...................275
4 Carathéodory numbers for measures supported on algebraic varieties ...............278
5 Lower bounds on the Carathéodory number .............................283
Communicated by Andreas Thom.
BPhilipp J. di Dio
Mario Kummer
1Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin,
Germany
2Technische Universität Dresden, Fakultät Mathematik, Institut für Geometrie, Zellescher Weg 12-14,
01062 Dresden, Germany
123
268 P. J. di Dio, M. Kummer
6 Hankel matrices and flat extension ..................................287
References ................................................289
1 Introduction
The theory of (truncated) moment sequences is a field of diverse applications and
connections to numerous other mathematical fields, see e.g. [1,22,29–31,33,34,36,38,
48,50,52], and references therein. For more on recent advances in the reconstruction
of measures from moments see e.g. [6,10,11,14,20,21,23,26,35,40,41], and references
therein.
A crucial fact in the theory of truncated moment sequences is the Richter (Richter–
Rogosinski–Rosenbloom) Theorem [43–45] which states that every truncated moment
sequence is a convex combination of finitely many Dirac measures, see also Theo-
rem 2.2. The Carathéodory number is the minimal number Nsuch that every truncated
moment sequence (with fixed truncation) is a sum of Natoms, i.e., Dirac measures.
It has been studied in several contexts but in most cases the precise value of the
Carathéodory number is not known [15,16,32,39,42,43,46,53].
In this work we proceed the study of Carathéodory numbers. We treat moment
sequences with small gaps (see Sect. 3), moment sequences of measures supported
on algebraic varieties (Sect. 4), and the multidimensional polynomial case on Rnand
[0,1]n(Sect. 5). For moment functionals with small gaps we find explicit lower and
upper bounds for dimension n=1 based on Descartes’ rule of signs, see Theorem 3.7.
For moment functionals L:R[X]≤2d→Ron polynomial functions on an algebraic
set X⊂Rnand for sufficiently large d, Theorem 4.5 yields an upper bound of
P(2d)−1 and a lower bound of
P(2d)−k·P(d)+k
2,
where Pis the Hilbert polynomial and kthe dimension of X. In the case X=Rnand
L:R[x1,...,xn]≤2d→R, this gives the lower bound
n+2d
n−n·n+d
n+n
2
(Theorem 5.2). We obtain similar bounds for odd degrees and the case X=[0,1]nin
Sect. 5. In Sect. 6we discuss implications of these bounds, when n→∞and d→∞.
We show that there are moment functionals L:R[x1,...,xn]≤2d→Rthat behave
as bad as possible under flat extensions, see Theorem 6.2 for the precise statement. For
literature on flat extensions in this context see [8,9,36,48] and the references therein.
123
Carathéodory numbers from Hilbert functions 269
2 Preliminaries
2.1 Truncated moment problem
Let Abe a (finite dimensional) real vector space of measurable functions on a mea-
surable space (X,A). Denote by L:A→Ra continuous linear functional. If there
is a (positive) measure μon (X,A)such that
L(a)=X
a(x)dμ(x), for all a∈A,(1)
then Lis called a moment functional. If Ais finite dimensional, it is a truncated
moment functional. By A={a1,...,am}we denote a basis of the m-dimensional real
vector space Aand by
si:= L(ai)
the ai-th (or simply i-th) moment of L(or μfor a μas in (1)). Given a sequence
s=(s1,...,sm)∈Rmwe define the Riesz functional Lsby setting Ls(ai)=si
for all i=1,...,mand extending it linearly to A, i.e., the Riesz functional induces
a bijection between moment sequences s=(s1,...,sm)and moment functionals
L=Ls.ByMAwe denote the set of all measures on (X,A)such that all a∈Aare
integrable and by MA(s)or MA(L)we denote all representing measures of the moment
sequence sresp. moment functional L. Even though moment sequences and moment
functionals are the same, when we apply techniques from algebraic geometry it is
easier to work with moment functionals L:A→Ron e.g. A=R[x1,...,xn]≤2d
or R[X]≤2dwhile when we work with Hankel matrices it is easier to work with
moment sequences sinafixedbasisAof A. Since the polynomials R[x1,...,xn]≤2d
are of special importance, we denote by
An,d:= {xα|α∈Nn
0∧|α|=α1+···+αm≤d}
the monomial basis, where we have xα=xα1
1···xαn
nwith α=(α1,...,α
n)∈Nn
0.
On Nn
0we work with the partial order α=(α1,...,α
n)≤β=(β1,...,β
n)if
αi≤βifor all i=1,...,n.
Definition 2.1 Let A={a1,...,am}be a basis of the finite dimensional vector space
Aof measurable functions on the measurable space (X,A). We define sAby
sA:X→Rm,x→ sA(x):= ⎛
⎜
⎝
a1(x)
.
.
.
am(x)
⎞
⎟
⎠.
Of course, sA(x)is the moment sequence of the Dirac δxmeasure and the cor-
responding moment functional is the point evaluation lxwith lx(a):= a(x).Bya
123
270 P. J. di Dio, M. Kummer
measure we always mean a positive measure unless it is explicitly denoted as a signed
measure.
The fundamental theorem in the theory of truncated moments is the following.
Theorem 2.2 (Richter Theorem [43, Satz 11]) Let A={a1,...,am},m ∈N,be
finitely many measurable functions on a measurable space (X,A). Then every moment
sequence s ∈SAhas a k-atomic representing measure
s=
k
i=1
ci·sA(xi)
with k ≤m, c1,...,ck>0, and x1,...,xk∈X.
The theorem can also be called Richter–Rogosinski–Rosenbloom Theorem [43–
45], see the discussion after Example 20 in [15] for more details. That every truncated
moment sequence has a k-atomic representing measure ensures that the Carathéodory
number CAis well-defined.
Definition 2.3 Let A={a1,...,am}be linearly independent measurable functions on
a measurable space (X,A).Fors∈SAwe define the Carathéodory number CA(s)of
sby
CA(s):= min{k∈N0|∃μ∈MA(s)k-atomic}.
We define the Carathéodory number CAof SAby
CA:= max
s∈SA
CA(s).
The same definition holds for moment functionals L:A→R.
The following theorem turns out to be a convenient tool for proving lower bounds
on the Carathéodory number CA.
Theorem 2.4 ([16, Thm. 18]) Let A={a1,...,am}be measurable functions on a
measurable space (X,A),s∈SA, and a ∈Awith a ≥0on X,Z(a)={x1,...,xk}
and Ls(a)=0. Then
CA≥CA(s)=dim lin {sA(xi)|i=1,...,k}.
Remark 2.5 Note that in Theorem 2.4 it is crucial that the zero set of ais finite: Take
a=0 and X=Rnfor a simple example where the statement fails when the zero set
is not finite.
It is well-known that in general not every sequence s∈Rmor linear functional
L:A→Rhas a positive representing measure. But of course it always has a signed
k-atomic representing measure with k≤m.
123
Carathéodory numbers from Hilbert functions 271
Lemma 2.6 ([15, Prop. 12]) Let A={a1,...,am}be a basis of the finite dimensional
space Aof measurable functions on a measurable space (X,A). There exist points
x1,...,xm∈Xsuch that every vector s ∈Rmhas a signed k-atomic representing
measure μwith k ≤m and all atoms are from {x1,...,xm}, i.e., every functional
L:A→Ris the linear combination L =c1lx1+···+cmlxm,c
i∈R.
It is well-known that in dimension n=1 the atom positions xiof a moment
sequence can be calculated from the generalized eigenvalue problem, see e.g. [24]. To
formulate this and other results we introduce the following shift.
Definition 2.7 Let n,d∈Nand s=(sα)α∈Nn
0:|α|≤d.Forβ∈Nn
0with |β|≤dwe
define Mβs:= (Mβsα)α∈Nn
0:|α+β|≤dby Mβsα:= sα+β, i.e., (MβL)(p)=L(xβ·p).
For a space Aof measurable functions with basis A={a1,a2...}the Hankel matrix
Hd(L)of a linear functional L:A2→Ris given by Hd(L)=(L(aiaj))d
i,j=1.The
atom positions of a truncated moment sequence s(resp. moment functional L)are
then determined by the following result from a generalized eigenvalue problem.
Lemma 2.8 Let n,d∈N,X=C, and s =(s0,s1,...,s2d+1)∈R2d+2with
s=
k
i=1
ci·sA1,2d+1(zi)
for some zi∈C,c
i∈C, and k ≤d. Then the ziare unique and are the eigenvalues
of the generalized eigenvalue problem
Hd(M1s)vi=ziHd(s)vi.(2)
Proof That the ziare the eigenvalues of (2) and therefore uniqueness follows from
Hd(s)=(sA1,d(z1),...,sA1,d(zk)) ·diag (c1,...,ck)·(sA1,d(z1), . . . , sA1,d(zk))T
and
Hd(M1s)
=(sA1,d(z1),...,sA1,d(zk)) ·diag (c1z1,...,ckzk)·(sA1,d(z1),...,sA1,d(zk))T.
We gave here only the 1-dimensional formulation, but a similar result holds also for
n>1. But as seen from the Carathéodory number and the flat extension in Sects. 5and
6, the size of the Hankel matrix of the flat extension can be very large. For numerical
reasons it is therefore advisable to reduce n-dimensional problems to 1-dimensional
problems.
123
Loading more pages...