ANZIAM J.
47 (2006), 451475

Sigmoidal cosine series on the interval


Abstract

We construct a set of functions, say,
composed of a cosine function and a sigmoidal
transformation
of order
. The present functions are orthonormal with
respect to a proper weight function on the
interval . It is proven that if a function
is continuous and piecewise smooth on
then its series expansion based on
converges uniformly to
so long as the order of the sigmoidal
transformation employed is
. Owing to the variational feature of
according to the value of
r, one can expect improvement of the traditional
Fourier series approximation for a function on a
finite interval. Several numerical examples show
the efficiency of the present series expansion
in comparison with the Fourier series expansion.

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