The uniqueness of everywhere convergent multiple Haar series (Q1851500)

The paper deals with the double Haar series \[ \sum\limits_{n=1}^{+\infty} \sum\limits_{m=1}^{+\infty} a_{n,m}\chi_{n,m}(x,y), \] where \(\chi_{n,m}(x,y) = \chi_n(x)\chi_m(y)\). Under quite general assumptions a theorem on monotonicity is proved. By means of this theorem a dyadic integral of Perron type is constructed and it is shown that the everywhere convergent Haar series is the Fourier series of its sum with respect to the constructed integral.