Abstract:
We prove that for arbitrary $\varepsilon>0$ there exists a sequence of positive integers $\{n_k\}$ such that a) the system $\{\cos n_kX,\sin n_kX\}$ is a basis with respect to the $C[-\pi,\pi]$ norm in the closure of its linear hull, and b) a continuous function $f(x)$ belonging to the closure of the linear hull of the system can be found such that its Fourier coefficients $a_n$ and $b_n$ satisfy the relation
$$
\sum{n=1}^\infty|a_n|^{2-\varepsilon}+|b_n|^{2-\varepsilon}=\infty.
$$
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