F. and M. Riesz theorem
Encyclopedia
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
dθ can be detected by means of Fourier coefficients.
More precisely, it states that if the Fourier-Stieltjes coefficients of
satisfy
for all
,
then μ is absolutely continuous with respect to dθ.
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
H1.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
and Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...
, on analytic measures. It states that for a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
μ on the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, any part of μ that is not absolutely continuous with respect to the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
dθ can be detected by means of Fourier coefficients.
More precisely, it states that if the Fourier-Stieltjes coefficients of
satisfy
for all
,then μ is absolutely continuous with respect to dθ.
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
Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...
and the existence of boundary values for the Hardy space
Hardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
H1.