In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums
to be dense in a weighted L2 space on the real line. It was discovered by Mark Krein in the 1940s.[1] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem.[2][3]
^Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65 (1–3): 1–3, 27–55. doi:10.1016/0377-0427(95)00099-2. MR 1379118.
^Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.
December 18, 2023
krein, condition, mathematical, analysis, provides, necessary, sufficient, condition, exponential, sums, displaystyle, left, lambda, quad, mathbb, lambda, right, dense, weighted, space, real, line, discovered, mark, krein, 1940s, corollary, also, called, provi. In mathematical analysis Krein s condition provides a necessary and sufficient condition for exponential sums k 1 n a k exp i l k x a k C l k 0 displaystyle left sum k 1 n a k exp i lambda k x quad a k in mathbb C lambda k geq 0 right to be dense in a weighted L2 space on the real line It was discovered by Mark Krein in the 1940s 1 A corollary also called Krein s condition provides a sufficient condition for the indeterminacy of the moment problem 2 3 Contents 1 Statement 2 Indeterminacy of the moment problem 2 1 Example 3 ReferencesStatement editLet m be an absolutely continuous measure on the real line dm x f x dx The exponential sums k 1 n a k exp i l k x a k C l k 0 displaystyle sum k 1 n a k exp i lambda k x quad a k in mathbb C lambda k geq 0 nbsp are dense in L2 m if and only if ln f x 1 x 2 d x displaystyle int infty infty frac ln f x 1 x 2 dx infty nbsp Indeterminacy of the moment problem editLet m be as above assume that all the moments m n x n d m x n 0 1 2 displaystyle m n int infty infty x n d mu x quad n 0 1 2 ldots nbsp of m are finite If ln f x 1 x 2 d x lt displaystyle int infty infty frac ln f x 1 x 2 dx lt infty nbsp holds then the Hamburger moment problem for m is indeterminate that is there exists another measure n m on R such that m n x n d n x n 0 1 2 displaystyle m n int infty infty x n d nu x quad n 0 1 2 ldots nbsp This can be derived from the only if part of Krein s theorem above 4 Example edit Let f x 1 p exp ln 2 x displaystyle f x frac 1 sqrt pi exp left ln 2 x right nbsp the measure dm x f x dx is called the Stieltjes Wigert measure Since ln f x 1 x 2 d x ln 2 x ln p 1 x 2 d x lt displaystyle int infty infty frac ln f x 1 x 2 dx int infty infty frac ln 2 x ln sqrt pi 1 x 2 dx lt infty nbsp the Hamburger moment problem for m is indeterminate References edit Krein M G 1945 On an extrapolation problem due to Kolmogorov Doklady Akademii Nauk SSSR 46 306 309 Stoyanov J 2001 1994 Krein condition Encyclopedia of Mathematics EMS Press Berg Ch 1995 Indeterminate moment problems and the theory of entire functions J Comput Appl Math 65 1 3 1 3 27 55 doi 10 1016 0377 0427 95 00099 2 MR 1379118 Akhiezer N I 1965 The Classical Moment Problem and Some Related Questions in Analysis Oliver amp Boyd Retrieved from https en wikipedia org w index php title Krein 27s condition amp oldid 1013817184, wikipedia, wiki, book, books, library,