View source for Complex moment problem, truncated

One of the [[Interpolation|interpolation]] problems in the complex domain.

Given a doubly indexed finite sequence of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201701.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201702.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201704.png" />, the truncated complex moment problem entails finding a positive [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201705.png" /> supported in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201706.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201707.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201708.png" /> is a truncated moment sequence (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c1201709.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017010.png" /> is a representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017011.png" />. The truncated complex moment problem serves as a prototype for several other moment problems to which it is closely related: the full complex moment problem prescribes moments of all orders, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017012.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017014.png" />-moment problem (truncated or full) prescribes a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017015.png" /> which is to contain the support of the representing measure [[#References|[a26]]]; and the multi-dimensional moment problem extends each of these problems to measures supported in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017016.png" /> [[#References|[a14]]]; moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017017.png" />-dimensional complex moment problem is equivalent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017018.png" />-dimensional real moment problem [[#References|[a8]]]. All of these problems generalize classical power moment problems on the real line, whose study was initiated by Th.J. Stieltjes (1894), H. Hamburger (1920–1921), F. Hausdorff (1923), and M. Riesz (1923) (cf. also [[Moment problem|Moment problem]] and [[#References|[a1]]], [[#References|[a27]]]).

The truncated complex moment problem is also related to subnormal operator theory [[#References|[a24]]], [[#References|[a29]]], [[#References|[a31]]], polynomial hyponormality [[#References|[a11]]], and joint hyponormality [[#References|[a32]]], [[#References|[a33]]] (cf. also [[Semi-normal operator|Semi-normal operator]]). Indeed, A. Atzmon [[#References|[a2]]] used subnormal operator theory to solve the full complex moment problem for the disc, and M. Putinar [[#References|[a18]]] found a related but different solution to the disc problem based on hyponormal operator theory. More generally, K. Schmüdgen [[#References|[a26]]] used an approach based on operator theory and semi-algebraic geometry to obtain the following existence theorem for representing measures [[#References|[a26]]] in the multi-dimensional full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017019.png" />-moment problem for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017020.png" /> is compact and semi-algebraic; this result encompasses several previously known special cases (cf. [[#References|[a4]]], [[#References|[a5]]], [[#References|[a15]]], [[#References|[a17]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017021.png" /> denote the multi-shift operator on multi-sequences and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017022.png" /> be a finite subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017023.png" />. Suppose that the [[Semi-algebraic set|semi-algebraic set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017024.png" /> is compact. Then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017025.png" />-dimensional full (real) moment sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017026.png" /> has a representing measure supported in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017027.png" /> if and only if the quadratic forms associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017029.png" /> are positive semi-definite (for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017030.png" /> that is a product of distinct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017031.png" />).

For general closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017032.png" />, the full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017033.png" />-moment problem continues (1998) to defy a complete solution. Hamburger's classical theorem (1920) gives necessary and sufficient conditions for the solvability of the full moment problem on the real line, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017034.png" />: A real sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017035.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017036.png" /> has a representing measure supported in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017037.png" /> if and only if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017038.png" />, the Hankel matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017039.png" /> is positive semi-definite (cf. also [[Nehari extension problem|Nehari extension problem]]; [[Synthesis problems|Synthesis problems]]). Hamburger's theorem serves as a prototype for much of moment theory, because it provides a concrete criterion closely related to the moments. Nevertheless, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017040.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017041.png" />), positivity alone is not sufficient to imply the existence of a representing measure [[#References|[a3]]], [[#References|[a14]]], [[#References|[a25]]] and a concrete condition for solvability of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017042.png" /> full moment problem (including solvability of the full complex moment problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017043.png" />) remains unknown (to date, 1999, perhaps the most definitive and comprehensive treatments of the full multi-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017044.png" />-moment problem can be found in [[#References|[a38]]], [[#References|[a39]]]).

In a different direction, M. Riesz (1923) proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017045.png" /> (as above) has a representing measure supported in a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017046.png" /> if and only if whenever a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017047.png" /> (with complex coefficients) is non-negative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017049.png" />. E.K. Haviland (1935, [[#References|[a16]]]) subsequently extended this result to the multi-variable full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017050.png" />-moment problem. Although Riesz' theorem solves the full moment problem in principle, it is very difficult to verify the Riesz criterion for a particular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017051.png" /> unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017052.png" /> is a half-line (the case studied by Stieltjes), an interval (the case studied by Hausdorff) or, as in Schmüdgen's theorem, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017053.png" /> is compact and semi-algebraic. The intractability of the Riesz–Haviland criterion is related to lack of an adequate structure theory for multi-variable polynomials that are non-negative on a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017054.png" /> [[#References|[a5]]], [[#References|[a14]]], [[#References|[a20]]], [[#References|[a22]]]; in particular, D. Hilbert (1888) established the existence of a polynomial, non-negative on the real plane, that cannot be represented as a sum of squares of polynomials (cf. [[#References|[a3]]], [[#References|[a14]]], [[#References|[a25]]]).

Because a truncated moment problem is finite in nature, one expects that in cases where a truncated moment problem is solvable, it should be possible to explicitly construct finitely atomic representing measures by elementary methods. (See below for such a construction for the truncated complex moment problem.) From this point of view, the multi-variable truncated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017055.png" />-moment problem subsumes the multi-variable quadrature problem of numerical analysis (cf. [[#References|[a6]]], [[#References|[a13]]], [[#References|[a23]]], [[#References|[a31]]]). In addition, J. Stochel [[#References|[a28]]] has proven that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017056.png" /> is a multi-variable full moment sequence, and if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017057.png" /> the truncated sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017058.png" /> has a representing measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017059.png" /> supported in a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017060.png" />, then some subsequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017061.png" /> converges (in an appropriate [[Weak topology|weak topology]]) to a representing measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017062.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017063.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017064.png" />. Thus, a complete solution of the truncated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017065.png" />-moment problem would imply a solution to the full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017066.png" />-moment problem.

Truncated multi-variable moment problems can be analyzed via the positivity and extension properties of the associated moment matrices [[#References|[a7]]], [[#References|[a8]]]. For the truncated complex moment problem, one associates to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017067.png" /> the moment matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017068.png" />, with rows and columns indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017069.png" />, as follows: the entry in row <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017070.png" /> and column <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017071.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017072.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017073.png" /> is a representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017075.png" /> (the set of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017076.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017077.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017078.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017079.png" /> denotes the coefficient vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017080.png" /> with respect to the above lexicographic ordering of the monomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017081.png" />. In particular, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017082.png" /> is positive semi-definite and that the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017083.png" /> contains at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017084.png" /> points [[#References|[a8]]] (cf. [[#References|[a30]]]).

It can be proven [[#References|[a8]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017085.png" /> has a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017086.png" />-atomic (minimal) representing measure if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017088.png" /> admits an extension to a moment matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017089.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017090.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017091.png" /> admits such a flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017092.png" /> (i.e. an extension that preserves rank), then there is a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017093.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017094.png" /> (the column space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017095.png" />). It can be shown [[#References|[a8]]], Chap. 5, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017096.png" /> then admits unique successive flat (positive) extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017098.png" /> is determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c12017099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170100.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170101.png" />). The resulting infinite moment matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170102.png" /> induces a semi-inner product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170103.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170104.png" />. The space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170105.png" /></td> </tr></table>

is an [[Ideal|ideal]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170107.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170108.png" />-dimensional Hilbert space on which the multiplication operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170109.png" /> is normal [[#References|[a8]]], Chap. 4. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170110.png" /> (cf. also [[Spectrum of a matrix|Spectrum of a matrix]]) then provides the support for the unique (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170111.png" />-atomic) representing measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170112.png" /> associated with the flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170113.png" />.

To explicitly construct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170114.png" />, note that since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170115.png" />, there is a linear relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170116.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170117.png" /> (or, equivalently, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170118.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170119.png" /> [[#References|[a12]]]). The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170120.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170121.png" /> distinct complex roots, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170122.png" />, which provide the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170123.png" />, and the densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170124.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170125.png" />, are uniquely determined by the Vandermonde equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170126.png" /></td> </tr></table>

[[#References|[a8]]], Chap. 4.

Results of [[#References|[a10]]] and [[#References|[a23]]] imply that the most general finitely atomic representing measures for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170127.png" /> correspond to positive, finite-rank moment matrix extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170128.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170129.png" />. Such an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170130.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170131.png" /> admits a positive extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170132.png" />, which in turn admits a flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170133.png" /> [[#References|[a10]]]. Examples for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170134.png" /> is required are provided in [[#References|[a37]]]. On the other hand, examples in [[#References|[a3]]], [[#References|[a25]]] imply that a positive, infinite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170135.png" /> need not correspond to any representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170136.png" />.

The preceding results suggest the following flat extension problem [[#References|[a9]]], [[#References|[a10]]]: under what conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170137.png" /> does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170138.png" /> admit a flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170139.png" />? Among the necessary conditions for a flat extension is the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170140.png" /> be recursively generated, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170142.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170143.png" />. Although not every recursively generated positive moment matrix admits a flat extension (or even a representing measure [[#References|[a10]]], [[#References|[a36]]]), several positive results are known:

i) [[#References|[a8]]] If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170144.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170145.png" /> admits a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170146.png" /> atomic representing measure.

ii) [[#References|[a9]]] If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170147.png" /> is recursively generated and if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170148.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170149.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170150.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170151.png" /> admits infinitely many flat extensions, each corresponding to a distinct rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170152.png" />-atomic (minimal) representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170153.png" />.

iii) [[#References|[a9]]] If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170154.png" /> is recursively generated and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170155.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170156.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170157.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170158.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170159.png" /> admits a unique flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170160.png" />.

The preceding approach can be extended to truncated moment problems in any number of real or complex variables; to do this one defines moment matrices subordinate to lexicographic orderings of the variables [[#References|[a8]]]. In the case of one real variable, such moment matrices are the familiar Hankel matrices, and the theory subsumes the truncated moment problems of Stieltjes, Hamburger, and Hausdorff [[#References|[a7]]] (cf. also [[Moment problem|Moment problem]]).

A refinement of the moment matrix technique also leads to an analogue of Schmüdgen's theorem for minimal representing measures in the truncated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170161.png" />-moment problem for semi-algebraic sets. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170163.png" />, and a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170164.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170165.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170166.png" />, there exists a unique matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170167.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170168.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170169.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170170.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170171.png" /> may be expressed as a linear combination of compressions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170172.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170173.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170174.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170175.png" />. There exists [[#References|[a34]]] a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170176.png" />-atomic (minimal) representing measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170177.png" /> supported in

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170178.png" /></td> </tr></table>

if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170179.png" /> admits a flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170180.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170181.png" /> (relative to the uniquely determined flat extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170182.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170183.png" />.

For additional recent (1999) results on the truncated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170184.png" />-moment problem, see [[#References|[a35]]], [[#References|[a36]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.J. Akhiezer, "The classical moment problem" , Hafner (1965) {{MR|0184042}} {{ZBL|0135.33803}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Atzmon, "A moment problem for positive measures on the unit disc" ''Pacific J. Math.'' , '''59''' (l975) pp. 317–325 {{MR|0402057}} {{ZBL|0319.44009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> C. Berg, T.P.R. Christensen, C.U. Jensen, "A remark on the multidimensional moment Problem" ''Math. Ann.'' , '''223''' (1979) pp. 163–169 {{MR|0543726}} {{ZBL|0416.46003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Berg, P.H. Maserick, "Polynomially positive definite sequences" ''Math. Ann.'' , '''259''' (1982) pp. 187–495 {{MR|0660043}} {{ZBL|0486.44004}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Cassier, "Probléme des moments sur un compact de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170185.png" /> et décomposition des polynômes a plusiers variables" ''J. Funct. Anal.'' , '''58''' (1984) pp. 254–266</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Cools, P. Rabinowitz, "Monomial cubature rules since `Stroud': a compilation" ''J. Comput. Appl. Math.'' , '''48''' (1993) pp. 309–326</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Recursiveness, positivity, and truncated moment problems" ''Houston J. Math.'' , '''17''' (1991) pp. 603–635 {{MR|1147276}} {{ZBL|0757.44006}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Solution of the truncated complex moment problem for flat data" ''Memoirs Amer. Math. Soc.'' , '''119''' (1996) {{MR|1303090}} {{ZBL|0876.30033}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Flat extensions of positive moment matrices: relations in analytic or conjugate terms" H. Bercovici (ed.) et al. (ed.) , ''Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics'' , Birkhäuser (1998) pp. 59–82 {{MR|1639649}} {{ZBL|0904.30020}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Flat extensions of positive moment matrices: recursively gnereated relations" ''Memoirs Amer. Math. Soc.'' , '''136''' (1998)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R. Curto, M. Putinar, "Nearly subnormal operators and moment problems" ''J. Funct. Anal.'' , '''115''' (1993) pp. 480–497 {{MR|1234402}} {{ZBL|0817.47026}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L. Fialkow, "Positivity, extensions and the truncated complex moment problem" ''Contemp. Math.'' , '''185''' (1995) pp. 133–150 {{MR|1332058}} {{ZBL|0830.44007}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> L. Fialkow, "Multivariable quadrature and extensions of moment matrices" ''Preprint'' (1996)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> B. Fuglede, "The multidimensional moment problem" ''Exposition Math.'' , '''1''' (1983) pp. 47–65 {{MR|0693807}} {{ZBL|0514.44006}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> F. Hausdorff, "Momentprobleme fur ein endliches Intervall" ''Math. Z.'' , '''16''' (1923) pp. 220–248 {{MR|1544592}} {{ZBL|49.0193.01}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> E.K. Haviland, "On the momentum problem for distributions in more than one dimension, I–II" ''Amer. J. Math.'' , '''57/58''' (1935/1936) pp. 562–568; 164–168</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J.L. McGregor, "Solvability criteria for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170186.png" />-dimensional moment problems" ''J. Approx. Th.'' , '''30''' (1980) pp. 315–333</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> M. Putinar, "A two-dimensional moment problem" ''J. Funct. Anal.'' , '''80''' (1988) pp. 1–8 {{MR|0960218}} {{ZBL|0668.47024}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> M. Putinar, "The L problem of moments in two dimensions" ''J. Funct. Anal.'' , '''94''' (1990) pp. 288–307 {{MR|1081646}} {{ZBL|0735.47008}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> M. Putinar, "Positive polynomials on compact semi-algebraic sets" ''Indiana Univ. Math. J.'' , '''42''' (1993) pp. 969–984 {{MR|1254128}} {{ZBL|0796.12002}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> M. Putinar, "Extremal solutions of the two-dimensional L-problem of moments" ''J. Funct. Anal.'' , '''136''' (1996) pp. 331–364 {{MR|1380658}} {{ZBL|0917.47014}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> M. Putinar, "Linear analysis of quadrature domains" ''Ark. Mat.'' , '''33''' (1995) pp. 357–376 {{MR|1373029}} {{ZBL|0892.47025}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> M. Putinar, "On Tchakaloff's theorem" ''preprint'' (1995)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> D. Sarason, "Moment problems and operators on Hilbert space" ''Moments in Math. — Proc. Sympos. Appl. Math.'' , '''37''' (1987) pp. 54–70 {{MR|921084}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> K. Schmüdgen, "An example of a positive polynomial which is not a sum of squares. A positive but not strongly positive functional" ''Math. Nachr.'' , '''88''' (1979) pp. 385–390 {{MR|543417}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> K. Schmüdgen, "The K-moment problem for semi-a1gebraic sets" ''Math. Ann.'' , '''289''' (1991) pp. 203–206</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> J. Shohat, J. Tamarkin, "The problem of moments" , ''Math. Surveys'' , '''I''' , Amer. Math. Soc. (1943) {{MR|0008438}} {{ZBL|0063.06973}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> J. Stochel, "private communication" ''private communication'' (1994)</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> J. Stochel, F. Szafraniec, "A characterisation of subnormal operators, spectral theory of linear operators and related topics" , Birkhäuser (1984) pp. 261–263</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> J. Stochel, F. Szafraniec, "Algebraic operators and moments on algebraic sets" ''Portug. Math.'' , '''51''' (1994) pp. 1–21 {{MR|1281954}} {{ZBL|0815.47058}} </TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> V. Tchakaloff, "Formules de cubatures mécaniques à coefficients non négatifs" ''Bull. Sci. Math.'' , '''81''' (1957) pp. 123–134 {{MR|0094632}} {{ZBL|0079.13908}} </TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Recursively generated weighted shifts and the subnormal completion problem" ''Integral Eq. Operator Th.'' , '''17''' (1993) pp. 202–216 {{MR|1233668}} {{ZBL|0804.47028}} </TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> R. Curto, L. Fialkow, "Recursively generated weighted shifts and the subnormal completion problem II" ''Integral Eq. Operator Th.'' , '''18''' (1994) pp. 369–426 {{MR|1265443}} {{ZBL|0807.47016}} </TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> R. Curto, L. Fialkow, "The truncated complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120170/c120170187.png" />-moment problem" ''Trans. Amer. Math. Soc.'' (to appear) {{MR|1303090}} {{ZBL|0876.30033}} </TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> R. Curto, L. Fialkow, "The quadratic moment problem for the unit circle and unit disk" ''Preprint'' (1999) {{MR|1799648}} {{ZBL|0979.44006}} </TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> R. Curto, L. Fialkow, "The quartic complex moment problem" ''Preprint'' (1999)</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> L. Fialkow, "Minimal representing measures arising from constant rank-increasing moment matrix extensions" ''J. Operator Th.'' (to appear)</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> M. Putinar, F. Vasilescu, "Solving moment problems by dimensional extension" ''Preprint'' (1998) {{MR|1709313}} {{MR|1679992}} {{ZBL|0939.44003}} {{ZBL|0931.44004}} </TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> J. Stochel, F. Szafraniec, "The complex moment problem and subnormality: A polar decomposition approach" ''J. Funct. Anal.'' , '''159''' (1998) pp. 432–491 {{MR|1658092}} {{MR|1709313}} {{ZBL|1048.47500}} </TD></TR></table>