On Jackson's theorem in \(L_ 2\) for price systems (Q1326051)

Given a sequence \(m= (m_ 0,m_ 1,\dots)\) with \(m_ k\in \mathbb{N}\) and \(m_ k\geq 2\) for \(k= 0,1,\dots\), consider the zero-dimensional, compact Abelian group \[ G_ m= \{x= (x_ 0,x_ 1,\dots): 0\leq x_ k\leq m_ k- 1, x_ k\in \mathbb{Z}_ +\} \] with the addition operation \(x+ y= ((x_ 0+ y_ 0)(\text{mod }m_ 0),(x_ 1+ y_ 1)(\text{mod }m_ 1),\dots)\). If \(M_ 0= 1\), \(M_ k= m_{k-1} M_{k-1}\), \(k\geq 1\), and \(n\in \mathbb{Z}_ +\) with \(n= \sum^ \infty_{k=0} n_ k M_ k\), where \(0\leq n_ k\leq m_ k-1\), we define, for \(x\in G_ m\), \(\phi_ n(x)= \exp(2\pi i\sum^ \infty_{k=0} n_ k x_ k/m_ k)\) and call \(\{\phi_ n\}^ \infty_{n=0}\) the Price system associated with \(m\). Let \(E_ n(f,G_ m)_ 2= \inf_{a_ k} \| f- \sum^{n-1}_{k=0} \phi_ k\|_ 2\), where \(\|\cdot\|_ 2\) denotes the \(L_ 2\) norm of \(G_ m\) with respect to the Haar measure on \(G_ m\), and let \[ \omega(\delta, f,G_ m)_ 2= \sup\left\{\| f(x+ t)- f(x)\|_ 2: | t|= \sum^ \infty_{i=0} t_ i/M_{i+1}< \delta\right\}. \] \textit{N. Ya. Vilenkin} and \textit{A. I. Rubinstein} [Izv. Vyssh. Uchebn. Zaved., Mat. 9(160), 3-9 (1975; Zbl 0322.43010)] have shown the (sharp) Jackson inequality \[ E_{M_ n}(f, G_ m)\leq {1\over \sqrt 2} \omega\left({\tau\over M_ n},f,G_ m\right),\tag{1} \] for \(\tau= 1\). In this paper, the author finds the smallest value \(\tau= \tau_{m,n}\) for which (1) holds for an arbitrary Price system. Then he asks for the sharp constant \[ K_ 2(\tau/M_ n, M_ n, G_ n)= \sup_{f\in L_ 2(G_ m)} {E_{M_ n}(f,G_ m)_ 2\over \omega(\tau/M_ n, f, G_ m)_ 2}, \] for \(1/m_ n< \tau< \tau_{m,n}\), and solves this problem for the Walsh system \(m\) (all \(m_ k= 2\)).