F. and M. Riesz theorem

F. and M. Riesz theorem

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 musatisfy hatmu_n=int_0^{2pi}{ m e}^{-in heta}frac{d heta}{2pi}=0,for all n<0, then μ is absolutely continuous with respect to "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"1.

References

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Riesz theorem — See:* F. and M. Riesz theorem * Riesz representation theorem * M. Riesz extension theorem * Riesz Thorin theorem * Riesz Fischer theoremFrigyes Riesz and Marcel Riesz were two brothers, both of whom were notable mathematicians …   Wikipedia

  • Frigyes Riesz — Infobox Scientist name = Frigyes Riesz box width = image width = caption = birth date = 1880 01 22 birth place = Győr, Hungary (Austria Hungary) death date = death date and age|1956|2|28|1880|1|22 death place = Budapest, Hungary residence =… …   Wikipedia

  • Théorème de F. et M. Riesz — Pour les articles homonymes, voir Théorème de Riesz. En mathématiques, le théorème de F. et M. Riesz est un résultat des deux frères Frigyes Riesz et Marcel Riesz sur les mesures analytiques, selon lequel pour une mesure complexe (en) μ sur… …   Wikipédia en Français

  • Riesz representation theorem — There are several well known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. The Hilbert space representation theorem This theorem establishes an important connection between a …   Wikipedia

  • Riesz–Thorin theorem — In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof… …   Wikipedia

  • Riesz–Fischer theorem — In mathematics, the Riesz–Fischer theorem in real analysis refers to a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and… …   Wikipedia

  • Riesz-Thorin theorem — In mathematics, the Riesz Thorin theorem, often referred to as the Riesz Thorin Interpolation Theorem or the Riesz Thorin Convexity Theorem is a result about interpolation of operators . This should not be confused with somewhat different… …   Wikipedia

  • Riesz, Frigyes — ▪ Hungarian mathematician born Jan. 22, 1880, Györ, Austria Hungary [now in Hungary] died Feb. 28, 1956, Budapest, Hungary       Hungarian mathematician and pioneer of functional analysis, which has found important applications to mathematical… …   Universalium

  • Satz von Kolmogorow-Riesz — Der Satz von Kolmogorow Riesz (nach Andrei Nikolajewitsch Kolmogorow und Marcel Riesz) ist ein Lehrsatz aus dem mathematischen Teilgebiet der Funktionalanalysis, der ein Kompaktheitskriterium für Teilmengen von Lp Räumen darstellt. Dieser Satz… …   Deutsch Wikipedia

  • Marcel Riesz — Born November 16, 1886(1886 11 16) Győr, Austria Hungary Died September 4, 1969(1969 09 04 …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”