Van der Corput inequality

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

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Let   be a real or complex inner product space with inner product   and induced norm  . Suppose that   and that  . Then

 

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

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We start by noticing that for any   there exists   (real or complex) such that   and  . Then,

 
 
  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  .
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  • A blog post by Terence Tao on correlation transitivity, including the van der Corput inequality [1]