Chinchin inequality

The Chintschin inequality , named after Alexander Jakowlewitsch Chintschin , is an inequality from the mathematical branch of functional analysis . It compares sums of squares with p-norms of associated linear combinations of Rademacher functions . According to the French transcription of the name Chintschin, this inequality is often found under the name Khintchine inequality .

Definitions

Be it real or complex numbers. These can be combined into a vector , where for or stands. As an element of the Euclidean or unitary vector space, this vector has a length . ${\displaystyle c_{1},\ldots ,c_{n}}$ ${\displaystyle c=(c_{1},\ldots ,c_{n})\in \mathbb {K} ^{n}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \textstyle \|c\|_{2}=(|c_{1}|^{2}+\ldots +|c_{n}|^{2})^{1/2}}$

It is the Rademacher functions. Then one can form the linear combination with the selected numbers as coefficients and thus obtain a restricted function which is obviously a step function and therefore lies in every L ^p ([0,1]) , where . The Chinchin inequality compares the p-norm of this linear combination with the length of the vector . ${\displaystyle r_{n}:[0,1]\rightarrow \mathbb {R} ,t\mapsto \operatorname {sgn}(\sin(2^{n}\pi t))}$ ${\displaystyle c_{1}r_{1}+\ldots +c_{n}r_{n}}$ ${\displaystyle [0,1]\rightarrow \mathbb {R} }$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle c}$

Formulation of the inequality

There are constants for each , so that applies to all : ${\displaystyle p\in [1,\infty )}$ ${\displaystyle A_{p},B_{p}>0}$ ${\displaystyle c_{1},\ldots c_{n}\in \mathbb {K} }$

${\displaystyle A_{p}\cdot \|(c_{1},\ldots ,c_{n})\|_{2}\leq \left\|\sum _{k=1}^{n}c_{k}r_{k}\right\|_{p}\leq B_{p}\cdot \|(c_{1},\ldots ,c_{n})\|_{2}}$.

If you use the definitions of the standards, that means

${\displaystyle A_{p}\cdot \left(\sum _{k=1}^{n}|c_{k}|^{2}\right)^{\frac {1}{2}}\leq \left(\int _{0}^{1}\left|\sum _{k=1}^{n}c_{k}r_{k}(t)\right|^{p}\mathrm {d} t\right)^{\frac {1}{p}}\leq B_{p}\cdot \left(\sum _{k=1}^{n}|c_{k}|^{2}\right)^{\frac {1}{2}}}$.

Remarks

The full inequality can be found for the first time in John Edensor Littlewood , but special cases were published by Chintschin as early as 1923, which is why the inequality bears his name.

For the inequality is trivial, it is then even equality. The reason is that the Rademacher functions in the Hilbert space form an orthonormal system and therefore ${\displaystyle p=2}$ ${\displaystyle L^{2}([0,1])}$

${\displaystyle \left\|\sum _{k=1}^{n}c_{k}r_{k}\right\|_{2}^{2}=\sum _{k=1}^{n}|c_{k}|^{2}}$

applies.

Optimal constants

Functions (blue) and (red) ${\displaystyle p\mapsto A_{p}}$ ${\displaystyle p\mapsto B_{p}}$

The usual proofs of the Chinchin inequality, as found in the cited textbooks, are not particularly laborious, but only provide very rough estimates for the constants. The determination of the optimal constants is much more difficult, these were found by Uffe Haagerup , based on the preliminary work of Stanisław Szarek . Let it denote the gamma function and the solution of the equation ${\displaystyle \Gamma }$ ${\displaystyle p_{0}}$

${\displaystyle \Gamma \left({\frac {p+1}{2}}\right)={\frac {\sqrt {\pi }}{2}}}$, this means ${\displaystyle p_{0}=1{,}84742\ldots }$

The optimal constants for the Chinchin inequalities in real spaces are thus:

${\displaystyle A_{p}={\begin{cases}2^{{\frac {1}{2}}-{\frac {1}{p}}}&{\text{falls }}1\leq p\leq p_{0}\\2^{\frac {1}{2}}\cdot \left({\frac {\Gamma ({\frac {p+1}{2}})}{\sqrt {\pi }}}\right)^{\frac {1}{p}}&{\text{falls }}p_{0}<p<2\\1&{\text{falls }}2\leq p<\infty \end{cases}}}$

and

${\displaystyle B_{p}={\begin{cases}1&{\text{falls }}1\leq p\leq 2\\2^{\frac {1}{2}}\cdot \left({\frac {\Gamma ({\frac {p+1}{2}})}{\sqrt {\pi }}}\right)^{\frac {1}{p}}&{\text{falls }}2<p<\infty \end{cases}}}$

application

From the estimates of the Chinchin inequality one reads directly that the closed subspace generated by the Rademacher functions in is isomorphic to the sequence space of the square-summable sequences, i.e. every Banach space contains a closed subspace that is too isomorphic. After Pitt's theorem has none for this property. Therefore, for cannot be too isomorphic. In contrast, according to the Riesz-Fischer theorem for even an isometric isomorphism . ${\displaystyle L^{p}([0,1])}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle L^{p}([0,1])}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p\not =2}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p\not =2}$ ${\displaystyle L^{p}([0,1])}$ ${\displaystyle p=2}$ ${\displaystyle L^{2}([0,1])\cong \ell ^{2}}$

Kahane-Chinchin inequality

Another obvious consequence of the Chinchin inequality is that the different p -norms are equivalent on the subspace generated by the Rademacher functions . This was generalized as follows by Jean-Pierre Kahane to the so-called Kahane-Chinchin inequality . A Rademacher sequence is a sequence of independently and identically distributed random variables in a probability space with for all . The Rademacher functions on the probability space [0,1] with the Lebesgue measure obviously form such a sequence. ${\displaystyle (\varepsilon _{n})_{n}}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle P(\varepsilon _{n}=1)=P(\varepsilon _{n}=-1)=\textstyle {\frac {1}{2}}}$ ${\displaystyle n}$

For each there is a constant , so that for each Banach space and each finite sequence and Rademacher sequence the inequalities ${\displaystyle 1\leq p<\infty }$ ${\displaystyle C_{p}}$ ${\displaystyle X}$ ${\displaystyle x_{1},\ldots ,x_{n}\in X}$ ${\displaystyle (\varepsilon _{n})_{n}}$

${\displaystyle \mathrm {E} \left\|\sum _{k=1}^{n}\varepsilon _{k}(\cdot )x_{k}\right\|\leq \left(\mathrm {E} \left\|\sum _{k=1}^{n}\varepsilon _{k}(\cdot )x_{k}\right\|^{p}\right)^{\frac {1}{p}}\leq C_{p}\cdot \mathrm {E} \left\|\sum _{k=1}^{n}\varepsilon _{k}(\cdot )x_{k}\right\|}$

exist, where E stands for the formation of the expected value .

Individual evidence

1. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 2.24
2. ^ Joseph Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5 , theorem in Chapter VII
3. ^ JE Littlewood: On a certain bilinear form , Quart. J. Math. Oxford (1930) Volume 1, pages 164-174
4. ↑ A. Khintchine: About dyadic fractions , Math. Journal (1923), Volume 18, pages 109-116
5. ^ Uffe Haagerup: The best constants in the Khintchine inequality , Studia Mathematica (1981), Volume 70 No. 3, pages 231-283
6. ^ JS Szarek: On the best constant on the Khintchine inequality , Studia Mathematica (1976), volume 58, pages 197-218
7. ↑ JP Kahane: Sur les sommes vectorielles Σ ± u , CR Acad. Sci. Paris (1964), Volume 259, Pages 2577-2580
8. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory , Springer-Verlag (2006), ISBN 978-1-4419-2099-7 , Theorem 6.2.5