On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

• Autorzy: Robert E. Zink
• Języki publikacji: EN

Abstrakty

EN

In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^{p}[0,1]$, 1 ≤ p < ∞. Although perhaps not probable, the latter result would seem to be a plausible one, since a Schauder system is closed, in the classical sense, in each of the $L^{p}$-spaces. This closure condition is not a sufficient one, however, since a great variety of subsystems can be removed from a Schauder system without losing the closure property, but it is not always the case that the orthonormalizations of the residual systems thus obtained are Schauder bases for each of the $L^{p}$-spaces. In the present work, this situation is examined in some detail; a characterization of those subsystems whose orthonormalizations are Schauder bases for each of the spaces $L^{p}[0,1]$, 1 ≤ p < ∞, is given, and a class of examples is developed in order to demonstrate the sorts of difficulties that may be encountered.

Kategorie tematyczne

• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2002
• Tom: 92
• Numer: 1
• Strony: 97-110

Daty

wydano

2002

Twórcy

autor

Robert E. Zink

• Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.

Identyfikatory

DOI

10.4064/cm92-1-9