The Franklin System as Schauder Basis for L p μ [ 0, 1 ]
Copyright policy )The Franklin System as Schauder Basis for L p μ [ 0, 1 ]
Let \(\mu\) be a totally-finite Borel measure on [0,1]. According to a result of \textit{Krancberg} [Inst. Electron. Mashinostroeniya Trudy MIEM 24, 14-21 (1971)], if the Franklin system constitutes a Schauder basis for \(L^ p_{\mu}[0,1]\), for a given \(p\in [1,\infty)\), then \(\mu\) is absolutely continuous with respect to the Lebesgue measure, i.e. there exists a nonnegative Lebesgue measurable function W such that \(\mu (E)=\int_{E}W(t)dt\) for each Lebesgue measurable set E. The main result of this paper is the following: The Franklin system is a Schauder bases for \(L^ p_{\mu}[0,1]\) for some \(p\in [1,\infty)\) if and only if W satisfies the condition \(A_ p\) introduced by \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 165, 207-226 (1972; Zbl 0236.26016)].
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), Franklin system, totally-finite Borel measure, Schauder basis, Muckenhoupt condition, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), Franklin system, totally-finite Borel measure, Schauder basis, Muckenhoupt condition, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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