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 Incoming links: Hankel matrix

A Hankel matrix has entries constant on each anti-diagonal: with indices starting at zero,

H_{ij} = a_{i + j}

. For example,

H = a_{0} a_{1} a_{2} a_{1} a_{2} a_{3} a_{2} a_{3} a_{4} .

This differs from a Toeplitz matrix, whose entries are constant along diagonals. A square Hankel matrix is symmetric, although it need not be positive semidefinite.

An important example uses moments

a_{k} = \int x^{k} d μ (x)

of a positive measure on the real line, assuming the needed moments are finite. For a real coefficient vector

c

c^{T} Hc = \int (\sum_{i} c_{i} x^{i})^{2} d μ (x) \geq 0,

x = 2

gives

a_{k} = 2^{k}

and

H = (1, 2, 4)^{T} (1, 2, 4)

A moment problem asks whether specified numbers

m_{0}, m_{1}, \dots

arise from a positive Borel measure

μ

on a prescribed subset of the real line:

m_{n} = \int x^{n} d μ (x) .

Existence, support restrictions and uniqueness are separate parts of the question. A probability measure must have

m_{0} = 1

a

gives

m_{n} = a^{n}

, while the uniform probability measure on

[0, 1]

gives

m_{n} = 1/ (n + 1)

Positivity imposes concrete constraints. For every finite coefficient vector

c

\sum_{i, j = 0}^{N} \overline{c_{i}} c_{j} m_{i + j} = \int \sum_{j = 0}^{N} c_{j} x^{j}^{2} d μ (x) \geq 0,

so every associated Hankel matrix is positive semidefinite. On a fixed compact interval, the full moment sequence determines a finite measure uniquely: equality on polynomials extends to equality on continuous functions by the Weierstrass approximation theorem. A finite list of moments generally leaves many possible measures, which explains the distinction between moment reconstruction and finite-data fitting.