mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Krein's condition provides a necessary and sufficient condition for exponential sums :

\left\,

to be

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

in a weighted L₂ space on the real line. It was discovered by

Mark Krein Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of fu ...

in the 1940s. A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the

moment problem In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ''μ'' to the sequences of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) ...

Statement

Let ''μ'' be an

absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

on the real line, d''μ''(''x'') = ''f''(''x'') d''x''. The exponential sums :

\sum_^n a_k \exp(i \lambda_k x), 
\quad a_k \in \mathbb, \, \lambda_k \geq 0

are dense in ''L''₂(''μ'') if and only if :

\int_^\infty \frac \, dx = \infty.

Indeterminacy of the moment problem

Let ''μ'' be as above; assume that all the moments :

m_n = \int_^\infty x^n d\mu(x), \quad n = 0,1,2,\ldots

of ''μ'' are finite. If :

\int_^\infty \frac \, dx < \infty

holds, then the

Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by t ...

for ''μ'' is indeterminate; that is, there exists another measure ''ν'' ≠ ''μ'' on R such that :

m_n = \int_^\infty x^n \, d\nu(x), \quad n = 0,1,2,\ldots

This can be derived from the "only if" part of Krein's theorem above.

Example

Let :

f(x) = \frac \exp \left\;

the measure d''μ''(''x'') = ''f''(''x'') d''x'' is called the Stieltjes–Wigert measure. Since :

\int_^\infty \frac dx
 = \int_^\infty \frac \, dx < \infty,

the Hamburger moment problem for ''μ'' is indeterminate.

References

{{Reflist Theorems in analysis Theorems in approximation theory