en>Astrokid: /* Change of basis matrix */ the coordinates transform according to the conjugate of the matrix according to which the basis vectors transform. The matrix mentioned was correct, but it was incorrectly labelled as a transpose

2013-10-20T13:27:34Z

Change of basis matrix: the coordinates transform according to the conjugate of the matrix according to which the basis vectors transform. The matrix mentioned was correct, but it was incorrectly labelled as a transpose

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{{no footnotes|date=December 2013}}

In mathematics, '''Turán's method''' provides lower bounds for [[exponential sum]]s and complex [[power sum]]s. The method has been applied to problems in [[equidistribution]].

The method applies to sums of the form

:<math> s_\nu = \sum_{n=1}^N b_n z_n^\nu \ </math>

where the ''b'' and ''z'' are complex numbers and ''ν'' runs over a range of integers. There are two main results, depending on the size of the complex numbers ''z''.

==Turán's first theorem==
The first result applies to sums ''s''ν where <math>|z_n| \ge 1</math> for all ''n''. For any range of ''ν'' of length ''N'', say ''ν'' = ''M'' + 1, ..., ''M'' + ''N'', there is some ''ν'' with |''s''''ν''| at least ''c''(''M'', ''N'')|''s''0| where

:<math> c(M,N) = \left({ \sum_{k=0}^{N-1} \binom{M+k}{k} 2^k }\right)^{-1} \ . </math>

The sum here may be replaced by the weaker but simpler <math>\left({ \frac{N}{2e(M+N)} }\right)^{N-1}</math>.

We may deduce [[Fabry's gap theorem]] from this result.

==Turán's second theorem==
The second result applies to sums ''s''ν where <math>|z_n| \le 1</math> for all ''n''. Assume that the ''z'' are ordered in decreasing absolute value and scaled so that |''z''1| = 1. Then there is some ν with

:<math> |s_\nu| \ge 2 \left({ \frac{N}{8e(M+N)} }\right)^N \min_{1\le j\le N} \left\vert{\sum_{n=1}^j b_n }\right\vert \ . </math>

==See also==
* [[Turán's theorem]] in graph theory

==References==
* {{cite book | last=Montgomery | first=Hugh L. | authorlink=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}

{{DEFAULTSORT:Turan's method}}
[[Category:Exponentials]]
[[Category:Analytic number theory]]

Spherical basis - Revision history

en>Astrokid: /* Change of basis matrix */ the coordinates transform according to the conjugate of the matrix according to which the basis vectors transform. The matrix mentioned was correct, but it was incorrectly labelled as a transpose