Abstract
It is well known that if $${f \in L_{2}(0,1)}$$ is an arbitrary function ( $${{f(x) \nsim 0}, x \in [0,1]}$$ ) then its Fourier coefficients with respect to general orthonormal systems (ONS) may belong only to $${\ell_2}$$ . Thus in the general case it is impossible to estimate these coefficients by moduli of continuity or moduli of smoothness of the given functions. In the present paper conditions are found which should be satisfied by ONS so that the coefficients of some classes of functions can be estimated by modulus of continuity or modulus of smoothness of these functions.
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