In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence , does there exist a distribution function on the interval such that:[1]

In other words, an affirmative answer to the problems means that are the first n + 1 Fourier coefficients of some measure on .

Characterization

The trigonometric moment problem is solvable, that is, is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Hermitian Toeplitz matrix

with for ,

is positive semi-definite.[2]

The "only if" part of the claims can be verified by a direct calculation. We sketch an argument for the converse. The positive semidefinite matrix defines a sesquilinear product on , resulting in a Hilbert space

of dimensional at most n + 1. The Toeplitz structure of means that a "truncated" shift is a partial isometry on . More specifically, let be the standard basis of . Let and be subspaces generated by the equivalence classes respectively . Define an operator

by

Since

can be extended to a partial isometry acting on all of . Take a minimal unitary extension of , on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure on the unit circle such that for all integer k

For , the left hand side is

So

which is equivalent to

for some suitable measure .

Parametrization of solutions

The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry .

See also

Notes

References

  • Akhiezer, Naum I. (1965). The classical moment problem and some related questions in analysis. New York: Hafner Publishing Co. (translated from the Russian by N. Kemmer)
  • Akhiezer, N.I.; Kreĭn, M.G. (1962). Some Questions in the Theory of Moments. Translations of mathematical monographs. American Mathematical Society. ISBN 978-0-8218-1552-6.
  • Geronimus, J. (1946). "On the Trigonometric Moment Problem". Annals of Mathematics. 47 (4): 742–761. doi:10.2307/1969232. ISSN 0003-486X. JSTOR 1969232.
  • Schmüdgen, Konrad (2017). The Moment Problem. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN 978-3-319-64545-2. ISSN 0072-5285.
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