Some Problems of Convergence of General Fourier Series

Banach [1] proved that good differential properties of function do not guarantee the a.e. convergence Fourier series this with respect to general orthonormal systems (ONS). On other hand it is very well known a sufficient condition for an given by Menshov–Rademacher Theorem. The paper deals sequence positive numbers $$(d_{n})$$ such multiplying coefficients $$(C_{n}(f))$$ functions bounded variation these one obtains convergent form $$\sum_{n=1}^{\infty}d_{n}C_{n}(f)\varphi_{n}(x).$$ It established resulting conditions are best possible.