Paley's theorem for Hankel matrices via the Schur test

2015/05/07 by John J. F. Fournier, Fournier, John J. F., Bradley G. Wagner +1
Computer Science · Mathematics · #15A60 #42A55 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Matrix Theory and Algorithms #Primary 47B35 #Secondary 15A45 #math.FA #msc:15A45 #msc:15A60 #msc:42A55 #msc:47B35

paper · pdf · doi:10.48550/arxiv.1505.01760

16 pages

arxiv created 2015/05/07 · openalex publication_date 2015/05/07 · arxiv updated 2015/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Paley's theorem about lacunary coefficients of functions in the classical space H¹ on the unit circle is equivalent to the statement that certain Hankel matrices define bounded operators on ℓ² of the nonnegative integers. Since that statement reduces easily to the case where the entries in the matrix are all nonnegative, it must be provable by the Schur test. We give such proofs with interesting patterns in the vectors used in the test, and we recover the best constant in the main case. We use related ideas to reprove the characterization of Paley multipliers from H¹ to H².

Related

• The missing proof of Paley's theorem about lacunary coefficients
• Paley inequality for the Weyl transform and its applications
• Paley's inequality for nonabelian groups
• Hankel matrices acting on the Dirichlet space
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