F. Riesz Theorem

The theorem of F. and M. Riesz

Suppose that G is a locally compact abelian group and that P ⊂ G is a semigroup (satisfies P + P ⊂ P ) that is closed and satisfies P ∩ (−P ) = {0} and P ∪ (−P ) = G. We define a total order on G by x ≤ y when y − x ∈ P . We verify that this is indeed a total order. (We remark that nowhere in this do we show the significance of P being closed; but in this note we shall be speaking about discret...

The Riesz representation theorem

The Operator Fejér-Riesz Theorem

The Fejér-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrixand operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum’s operator generalization of the classical Fejér-Riesz theorem. Mathematics Subject Classification (2000). Primary 47A68; Secondary 60G25, 47A56, 47B35, 42A05, 32A70, 30E99.

The Kolmogorov–riesz Compactness Theorem

We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.

S ep 2 00 1 AN F . AND M . RIESZ THEOREM FOR PLANAR VECTOR FIELDS

We prove that solutions of the homogeneous equation Lu = 0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if Ω is an open subset of the plane with smooth boundary, u ∈ C 1 (Ω) satisfies Lu = 0 on Ω, has tempered growth at the boundary, and its weak boundary value is a measure µ, then µ is absolutely continuo...