ANZIAM J.
47 (2006), 451-475
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Sigmoidal cosine series on the interval
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Abstract
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We construct a set of functions, say,
![$\psi_n^{[r]}$](img_files_abstracts\2294-img0.png)
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 ![$[-1,1]$](img_files_abstracts\2294-img3.png) . It is proven that if a function
 is continuous and piecewise smooth on
![$[-1,1]$](img_files_abstracts\2294-img5.png) then its series expansion based on
![$\psi_n^{[r]}$](img_files_abstracts\2294-img6.png) converges uniformly to
 so long as the order of the sigmoidal
transformation employed is
 . Owing to the variational feature of
![$\psi_n^{[r]}$](img_files_abstracts\2294-img9.png) 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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Australian Mathematical Publishing Association Inc.
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©
Australian MS
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