Vol. 23, No. 2, 1967

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ISSN: 0030-8730
A characterization of restrictions of Fourier-Stieltjes transforms

Haskell Paul Rosenthal

Vol. 23 (1967), No. 2, 403–418
Abstract

The main result that we prove here is as follows: Let E be a Lebesgue measurable subset of R, the real line, and let φ be a bounded measurable function defined on E. Then the first of the following conditions implies the second: (1) There exists a constant K, so that

∑n
|   cjφ(xj)| ≦ K ∥P∥∞
j=1

for all trigonometric polynomials of the form P(y) = j=1ncjeixjy, where xj E for all 1 j n. (2) φ is E-almost everywhere a Stieltjes transform. Precisely, there exists a finite (complex Borel) measure μ, so that

            ∫ ∞  − ixy
φ(x) = ˆμ(x) = − ∞ e  dμ(y)

for almost all x E. Moreover, μ may be chosen such that μK, where K is the constant in (1). (μdenotes the total variation of μ.)

Mathematical Subject Classification
Primary: 42.25
Milestones
Received: 17 January 1967
Published: 1 November 1967
Authors
Haskell Paul Rosenthal