Weyl's inequality

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Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies

$|c-a/q|\le tq^{-2}$ ,

for some t greater than or equal to 1, then for any positive real number $\varepsilon$ one has

$\sum_{x=M+1}^{M+N}\exp(2\pi if(x))\in O\left(N^{1+\varepsilon}\left({t\over q}+{1\over N}+{t\over N^{k-1}}+{q\over N^k}\right)^{2^{1-k}}\right).$

This inequality will only be useful when

q < N k,

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as $\le N$ provides a better bound.

[edit] Weyl's inequality in matrix theory

In linear algebra, Weyl's Inequality is a theorem about the changes to eigenvalues of a Hermitian matrix that is perturbed. It is useful if we wish to know the eigenvalues of the Hermitian matrix H but there is an uncertainty about the entries of H. We let H be the exact matrix and P be a perturbation matrix that represents the uncertainty. The matrix we 'measure' is $M = H + P$ .

The theorem says that if M, H and P are all n by n Hermitian matrices, where M has eigenvalues

$\mu_1 \ge ... \ge \mu_n$