Singularities of carleman type for subsystems of a trigonometric system

We prove that for arbitrary ε>0 there exists a sequence of positive integers {n_k} such that a) the system { cos n_k X, sin n_k X} is a basis with respect to the C[-π, π] norm in the closure of its linear hull, and b) a continuous functionf(x) belonging to the closure of the linear hull of the system can be found such that its Fourier coefficientsa _n and b_n satisfy the relation $$\sum\nolimits_{n = 1}^\infty {\left| {a_n } \right|^{2 - \varepsilon } + \left| {b_n } \right|^{2 - \varepsilon } } = \infty $$ .     

Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on Vilenkin group $\mathfrak{G}$ . We consider a class of refinable step functions for which the mask m ₀(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$ and its modulus |m ₀(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$ . We show the sharpness of this result, too.     

On the Uniqueness Sets of Multiple Walsh Series for Convergence in Cubes

Mathematical Notes - Let $$G^d$$ be a power of the Cantor binary group $$G$$ . The uniqueness problem for a multiple Walsh series on a power of the binary group in the case of convergence in cubes...