The Jackson theorem in \(L_ 2\) for Chrestenson-Levy systems (Q1326082)

The author obtains exact values for constants in the Jackson inequalities (which compare the rate of best approximation of a function \(f\) by Vilenkin polynomials to its modulus of continuity) for multiplicative Vilenkin systems whose generating radices \(p_ 1,p_ 2,\dots\) satisfy \(p_ j\equiv p\) for some fixed integer \(p\). These results, which contain earlier results of \textit{V. I. Ivanov} [Mat. Zametki 53, No. 3, 37-50 (1993; Zbl 0811.41012)] for the Walsh case, i.e., the case \(p= 2\), are complete for \(p= 3\) and 4 (valid for all values of a parameter \(\tau\)) but only valid for \({1\over p}< \tau\leq {1\over p}+ {1\over p^ 2}\) for \(p> 4\).