Abstract
We present a unified theory of the p,p′ Fourier transformation Fp,p′, over any locally compact abelian group Γ for p ∈ [1,2], p′ := p/(p-1), based on (proper) Lebesgue integration with respect to a vector measure. This generalizes the case p = 2 settled in 1969, [8]. It transpires that for p ∈ (1,2), Lp (Γ) is only a proper subset of the domain of F p,p′.
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Masani, P. (1984). An Outline of the Fourier Transformation as Integration with Respect to a Vector Measure. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_40
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DOI: https://doi.org/10.1007/978-3-0348-5432-0_40
Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-5432-0
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