In mathematics, the van der Corput inequality is a corollary of the Cauchy–Schwarz inequality that is useful in the study of correlations among vectors, and hence random variables. It is also useful in the study of equidistributed sequences, for example in the Weyl equidistribution estimate. Loosely stated, the van der Corput inequality asserts that if a unit vector
in an inner product space
is strongly correlated with many unit vectors
, then many of the pairs
must be strongly correlated with each other. Here, the notion of correlation is made precise by the inner product of the space
: when the absolute value of
is close to
, then
and
are considered to be strongly correlated. (More generally, if the vectors involved are not unit vectors, then strong correlation means that
.)
Statement of the inequality[edit]
Let
be a real or complex inner product space with inner product
and induced norm
. Suppose that
and that
. Then
![{\displaystyle \displaystyle \left(\sum _{i=1}^{n}|\langle v,u_{i}\rangle |\right)^{2}\leq \sum _{i,j=1}^{n}|\langle u_{i},u_{j}\rangle |.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/227fbb176636cadea447a89806e5b7f0c0927ab1)
In terms of the correlation heuristic mentioned above, if
is strongly correlated with many unit vectors
, then the left-hand side of the inequality will be large, which then forces a significant proportion of the vectors
to be strongly correlated with one another.
Proof of the inequality[edit]
We start by noticing that for any
there exists
(real or complex) such that
and
. Then,
![{\displaystyle \left(\sum _{i=1}^{n}\left|\langle v,u_{i}\rangle \right|\right)^{2}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/14fc64ecb0c00687b80a7323f52af8b7af051006)
![{\displaystyle =\left(\sum _{i=1}^{n}\epsilon _{i}\langle v,u_{i}\rangle \right)^{2}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/187be7f31651a3c6c124d01ae077948300830641)
since the inner product is bilinear
by the Cauchy–Schwarz inequality
by the definition of the induced norm
since
is a unit vector and the inner product is bilinear
since
for all
.
External links[edit]
- A blog post by Terence Tao on correlation transitivity, including the van der Corput inequality [1]