F. And M. Riesz Theorem

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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 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 coefficients.More precisely, it states that if the Fourier-Stieltjes coefficients of <math>mu</math>satisfy*d*θ.

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>.

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- <math>hatmu_n=int_0^^frac=0, </math>

The original statements are rather different (see Zygmund,

- F. and M. Riesz,
*Über die Randwerte einer analytischen Funktion*, Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, (1916), pp. 27-44.

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