Abstract:
We present results describing some properties of the Fourier coefficients
of functions with respect to general orthonormal systems (ONS). We note that good
differential properties of the functions do not ensure the ‘good’ behaviour
of the Fourier coefficients (in the sense of convergence to zero) of these
functions with respect to general ONS. We find conditions on the functions
$\varphi_n(x)$ forming an ONS ($\varphi_n(x))$, $n=1,2,\dots$, for which the
series of Fourier coefficients of the functions $f(x)$, where $f'(x)\in V(0,1)$,
are absolutely convergent. We consider relationships between ONS, that is,
problems of absolute independence for orthonormal systems.
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