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Non-similarity of Walsh and trigonometric systems. (English) Zbl 0971.42018

It is proved that in \(L_p\) \((p\neq 2)\) the constants of equivalence between finite initial segments of the Walsh and trigonometric systems have power type growth. It follows from this result that the Walsh and trigonometric systems are equivalent in \(L_p\) only when \(p=2\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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