Editing Moment problem (section)

==Existence==

A sequence of numbers <math>m_n</math> is the sequence of moments of a measure <math>\mu</math> if and only if a certain positivity condition is fulfilled; namely, the [[Hankel matrices]] <math>H_n</math>,

:<math>(H_n)_{ij} = m_{i+j}\,,</math>

should be [[positive-definite matrix|positive semi-definite]]. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional <math> \Lambda</math> such that <math>\Lambda(x^n) = m_n</math> and  <math> \Lambda(f^2) \geq 0 </math> (non-negative for sum of squares of polynomials). Assume <math> \Lambda</math> can be  extended to <math> \mathbb{R}[x]^*</math>. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional <math> \Lambda</math> is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is <math> \Lambda(x^n) = \int_{-\infty}^{\infty} x^n d \mu</math>. A condition of similar form is necessary and sufficient for the existence of a measure <math>\mu</math> supported on a given interval <math>[a,b]</math>.

One way to prove these results is to consider the linear functional <math>\varphi</math> that sends a polynomial

:<math>P(x) = \sum_k a_k x^k </math>

to

:<math>\sum_k a_k m_k.</math>

If <math>m_k</math> are the moments of some measure <math>\mu</math> supported on <math>[a,b]</math>, then evidently

{{NumBlk|::|<math> \varphi(P) \ge 0</math> for any polynomial <math>P</math> that is non-negative on <math>[a,b]</math>.|{{EquationRef|1}}}}

Vice versa, if ({{EquationNote|1}}) holds, one can apply the [[M. Riesz extension theorem]] and extend <math>\varphi</math> to a functional on the [[Function space#Functional analysis|space of continuous functions with compact support]]  <math>C_c([a,b])</math>), so that

{{NumBlk|::|<math>\varphi(f) \ge 0</math> for any  <math>f \in C_c([a,b]),\;f\ge 0.</math>|{{EquationRef|2}}}}

By the [[Riesz representation theorem#The representation theorem for linear functionals on Cc.28X.29|Riesz representation theorem]], ({{EquationNote|2}}) holds iff there exists a measure <math>\mu</math> supported on <math>[a,b]</math>, such that

:<math> \varphi(f) = \int f \, d\mu</math>

for every <math>f \in C_c([a,b])</math>.

Thus the existence of the measure <math>\mu</math> is equivalent to ({{EquationNote|1}}). Using a representation theorem for positive polynomials on <math>[a,b]</math>, <!-- This is due to Riesz or Fejer (or maybe both); a ref. is needed (maybe Szego's book?) --> one can reformulate ({{EquationNote|1}}) as a condition on Hankel matrices.{{sfn|Shohat|Tamarkin|1943}}{{sfn|Kreĭn |Nudel′man|1977}}