In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick complex numbers , and add them together each multiplied by a random sign , then the expected value of its modulus, or the modulus it will be closest to on average, will be not too far off from .
Statement of theorem
for some constants depending only on (see Expected value for notation). The sharp values of the constants were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that when , and when .
Uses in analysis
The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let be a linear operator between two Lp spaces and , , with bounded norm , then one can use Khintchine's inequality to show that
for some constant depending only on and .
- Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
- Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
- Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.