https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=27.99.2.51formulasearchengine - User contributions [en]2026-05-22T18:07:25ZUser contributionsMediaWiki 1.43.0-wmf.28https://en.formulasearchengine.com/index.php?title=George_Secor&diff=11842George Secor2014-01-02T09:17:22Z<p>27.99.2.51: /* Further reading */ listed where article appeared</p>
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<div>In [[mathematics]], the '''F. and M. Riesz theorem''' is a result of the brothers [[Frigyes Riesz]] and [[Marcel Riesz]], on '''analytic measures'''. It states that for a [[measure (mathematics)|measure]] μ on the [[circle]], any part of μ that is not [[absolutely continuous]] with respect to the [[Lebesgue measure]] ''d''θ can be detected by means of [[Fourier coefficient]]s. <br />
More precisely, it states that if the Fourier-Stieltjes coefficients of <math>\mu</math><br />
satisfy<br />
:<math>\hat\mu_n=\int_0^{2\pi}{\rm e}^{-in\theta}\frac{d\mu(\theta)}{2\pi}=0,\ </math><br />
for all <math>n<0</math>, <br />
then μ is absolutely continuous with respect to ''d''θ.<br />
<br />
The original statements are rather different (see Zygmund, ''Trigonometric Series'', VII.8). The formulation here is as in Rudin, ''Real and Complex Analysis'', p.335. The proof given uses the [[Poisson kernel]] and the existence of boundary values for the [[Hardy space]] ''H''<sup>1</sup>.<br />
<br />
==References==<br />
*F. and M. Riesz, ''Über die Randwerte einer analytischen Funktion'', Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, (1916), pp. 27-44.<br />
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[[Category:Theorems in measure theory]]<br />
[[Category:Fourier series]]</div>27.99.2.51