An Introduction to Harmonic Analysis

Abstract

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Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group

“…Though this is probably a standard fact in Fourier analysis, we found a proof (in [8]) of only one direction.…”

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

“…Therefore, (11) implies that H a f ∈ VMOA. Now, if we assume that H a is compact on H 1 then it is also compact on VMOA, since H 1 is isomorphic to (VMOA) * .…”

Cyclicity of bicyclic operators and completeness of translates

“…We shall denote in the sequel the zero set of f by Z ( f ). We refer the reader to [30] or [33] for more on regular Banach algebras.…”

“…In what follows µ will denote an element of A ρ (T) * . The support of µ is well defined (see [30,Chap. 8]): it is the smallest closed subset K of T such that µ annihilates every function f ∈ A ρ (T) whose support is included in the complement of K .…”

“…Then by the regularity of the Banach algebra A ρ (T), there exists a function h ∈ Y ρ,0 which does not vanish on Γ . Let (h) denote the principal ideal generated by the function h. Since supp( f ) and Z (h) are disjoint, f belongs to the ideal (h) [30,p. 224,Corollary 5.7].…”