On partial sums of Fourier-Haar series with arbitrary dilation factor (Q1946438)

In this short communication, the author considers a ``classical Haar system with natural numbering'', recalling that for ``any continous 1-periodic function, its Fourier series with respect to a given system converges uniformly''. Denoting by \(\mathcal{C}\) the space of continous \(1\)-periodic functions, and, as usual, by \(\omega\left( f,t \right)\) the modulus of continuity of a function \(f\), a result from \textit{N. P. Khoroshko} [Ukr. Mat. Zh. 22, 705--712 (1970; Zbl 0216.39503)] asserts that \[ \sup_{f \in H\omega} \| f-S_{n}\left( f \right) \|_{\infty} = 2^{i} \int_{0}^{2^{-i}} \omega(t)\, dt;\;\; n=2^{i}+j, \] \[ H\omega = \{ f \in \mathcal{C}: \omega\left( f,t \right) \leq \omega(t),\, t \in \left[0,1\right]\}. \] The author points out that this result yields that in the well known inequality \[ \| f-S_{n}\left( f \right) \|_{\infty} \leq K \omega\left( f, \frac{1}{n} \right), \] the smaller value that can be achieved for the constant \(K\) is \(3/2\). Assuming a variable dilation factor, denoted by \(p\), the author studies the behaviour of the correspondent Haar systems, indicating interesting identities related to the speed of convergence of the Haar-Fourier series of a function \(f \in \mathcal{C}\), and also the following inequality: \[ \| f-S_{n}\left( f \right) \|_{\infty} \leq C \omega\left( f, \frac{1}{n} \right),\;\;\text{ where } C = \max\left\{ \frac{p+1}{2}, \max_{l=1,\dots,p-2} C_{l}\right\}, \] for a set of auxiliary quantities \(C_{l},\; l=1,\dots,p-2\), uniquely defined by an orthogonal matrix \(A\). Furthermore, the author shows that an analogous result to the one obtained by N. P. Khoroshko does not hold due to the fact that the next estimate is not valid in general. \[ \| f-S_{n}\left( f \right) \|_{\infty} = p^{i} \int_{0}^{p^{-i}} \omega\left( f, t \right)\, dt;\;\; n=p^{i}+j. \]