The Fourier coefficients of functions with bounded variation (Q2439381)

For several well-known complete orthonormal systems of functions in \(L^2([0,1])\) (trigonometric polynomials, Walsh, Haar), it is known that the Fourier coefficients of a function of bounded variation satisfy \[ \sum_{k=n}^\infty \, \left(c_k(f)\right)^2 \leq \frac{M_f}{\sqrt{n}} \] where \(M_f\) is a constant dependent on \(f\). However, this inequality is not true for all complete orthonormal systems of functions. The main results of this paper give necessary and sufficient conditions for a given complete orthonormal system to satisfy the above inequality. The conditions are related to the asymptotic growth of the expression \[ \max_{1 \leq i \leq n} \left| \int_0^{1/n} \, \sum_{k=n}^{n+p} c_k(f) \varphi_k(x)\, dx\right| \] with respect to \(n\), where \(\varphi_k\) is the \(k\)'th member of the orthonormal system of functions, and \(c_k(f)\) is the Fourier coefficient of \(f\) with respect to \(\varphi_k\).