Some properties of general Fourier coefficients of Lipschitz functions (Q822625)

Let \(r_k(t)\) be the Rademacher functions. Given an orthonormal system \(\{\varphi_n:\,n\ge 1\}\) on \([0,1]\), set \(H_n(x,t)= \sum_{k=1}^n \frac{\varphi_k(x)r_k(t)}{\sqrt k}\) and \(V_n(t) = \frac{1}{n}\sum_{i=1}^{n-1} \left\vert \int_0^{i/n}H_n(x,t) \text{d}x\right\vert \). The main results are the following.\par Theorem 1. \, Let \(\{\varphi_n:\,n\ge 1\}\) be an orthonormal system on \([0,1]\) such that for any \(t\in [0,1]\), \(\int_0^1 H_n(x,t)\text{d}x=O (1)\). If for an arbitrary fixed \(t\in[0,1]\), \(V_n(t)=O(1)\), then for any Lipschitz function \(f\) with \(\sup_{\vert x-y\vert \le h} \vert f(x)-f(y)\vert =O(h)\), there holds \[ \sum_{n=1}^{\infty} \frac{\vert C_n(f)\vert}{\sqrt n}<\infty,\] where \(C_n(f) = \int_0^1 f(x)\varphi_n(x)\text{d}x\).\par Theorem 2. \, Let \(\{\varphi_n:\,n\ge 1\}\) be an orthonormal system on \([0,1]\). If \(\limsup_{n\rightarrow\infty} V_n(t)=+\infty\) for some \(t\), then there exists a Lipschitz function \(g\) such that \[ \sum_{n=1}^{\infty} \frac{\vert C_n(g)\vert}{\sqrt n}=\infty.\]