F. and M. Riesz theorem

A finite complex measure on the unit circle with all Fourier coefficients vanishing on one side is absolutely continuous with respect to arc length. Its density is the boundary function of a Hardy-space function.

Connect

Fourier–Stieltjes transform: The sequence of Fourier coefficients obtained by integrating characters against a finite measure. The theorem's one-sided vanishing condition is imposed directly on these coefficients.
Unit circle: The set of complex numbers with modulus one, parametrized by an angle. It is the domain on which the measure and its Fourier coefficients are defined.
Frigyes Riesz: A Hungarian mathematician whose work shaped functional analysis, harmonic analysis, and measure theory. He proved the theorem's original result concerning analytic measures.
F. and M. Riesz theorem for analytic measures: The classical formulation stating that a finite measure with vanishing negative Fourier coefficients has an integrable density. It is the standard one-sided version of the theorem on the circle.
Hardy space: A space of analytic functions whose boundary values satisfy an integral norm bound. The absolutely continuous measure has a density represented by a Hardy-space boundary function.
Fourier coefficient: A scalar measuring a function or measure's component at a chosen frequency. The theorem's hypothesis specifies which indexed coefficients must vanish.
Marcel Riesz: A Hungarian mathematician known for contributions to analysis, including interpolation and harmonic analysis. His related work is commemorated alongside Frigyes Riesz in the theorem's name.
Singular measure: A measure concentrated on a set of zero measure under a specified reference measure. The theorem shows that the prescribed Fourier vanishing cannot coexist with a nonzero singular part.
Poisson kernel: A family of positive kernels that approximate the identity and extend boundary data harmonically into the disk. Poisson integrals connect the measure's Fourier coefficients to analytic functions inside the disk.
Haar measure: The translation-invariant measure on a compact group, normalized to have total mass one. On the circle, normalized Haar measure supplies the reference notion of arc length.