Coefficients of convergent multiple Haar series

It is well-known that if a multiple trigonometric series almost everywhere converges in the square or restricted rectangular sense to a finite function, then its coefficients grow slower than any exponential function. In this paper we prove the existence of a multiple Haar series that converges in the square or restricted rectangular sense to a finite function and contains a subsequence of coefficients that grows faster than any sequence defined an advance. Moreover, we show that for such series conditions of the Arutyunyan–Talalyan type can be violated at some points.