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Summary: NONUNIQUENESS FOR A BOUNDED, DIFFERENTIABLE
COMPLETE ORTHONORMAL SYSTEM IN L2[O,1]
J. MARSHALLASH ANDGANGWANG
Abstract. TheargumentofMftsegianand Ovsepjanis adaptedtoproducea
completeorthonormalsystemon [0,1]of uniformlyboundedfunctions,differen-
tiableon [13,1], and C~ on [0,1), for whichthe analogueof Cantor'suniqueness
theoremis false. Wealsoconstructa completeorthonormalsystemofC~ func-
tionswhichvanishto infiniteorderat bothendpoints.
1 Introduction
It was proved in 1870 by Georg Cantor that if a trigonometric series
1
7ao + (a. 2.,-,x + b. si.
'n.=l
sums to zero at every point x E [0, 1], then all the coefficients must be zero
[Z3]. Analogous theorems hold for a wide variety oforthonormal systems. These
include Sturm Liouville systems [HI, [Z1]; Jacobi polynomial systems [Z2], which
include ultraspherical polynomials [Kog], in particular Legendre polynomials [P 1];
and Bessel function systems [P2], [Z2]. Nevertheless, it has been known for a long
time that a general analogue of Cantor's theorem for all orthonormal systems
cannot be true. For example, the analogue of Cantor's theorem fails for the Haar
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