On generalized harmonic analysis

Abstract:

Motivated by Wiener’s work on generalized harmonic analysis, we consider the Marcinkiewicz space ${{\mathcal {M}}^p}({\textbf {R}})$ of functions of bounded upper average p power and the space ${{\mathcal {V}}^p}({\textbf {R}})$ of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that ${{\mathcal {V}}^p}({\textbf {R}})$, $1 < p < \infty$, is a Banach space. The proof depends on the result that each equivalence class in ${{\mathcal {V}}^p}({\textbf {R}})$ contains a representative in ${L^p}({\textbf {R}})$. This result, in turn, is based on Masani’s work on helixes in Banach spaces. Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace ${{\mathcal {W}}^2}({\textbf {R}})$ of ${{\mathcal {M}}^2}({\textbf {R}})$ consisting of functions of bounded average quadratic power, into the nonlinear subspace ${\mathcal {U}^2}({\textbf {R}})$ of ${{\mathcal {V}}^2}({\textbf {R}})$ consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener’s transformation W is actually an isomorphism from ${{\mathcal {M}}^2}({\textbf {R}})$ onto ${{\mathcal {V}}^2}({\textbf {R}})$. We also show by counterexamples that W is not an isometry on the closed subspace generated by ${{\mathcal {W}}^2}({\textbf {R}})$.