The construction of resonance curves in the author's monograph ‘Area, Lattice Points, and Exponential Sums’ is modified so that the resonance curves now have a differential equation, a functorial mapping property, and better approximation properties.

It is proved that a general functional equation of the Riemann type with multiple gamma factors has non-trivial solutions in the space of generalized Dirichlet series. Moreover, for a fixed functional equation, the space of such solutions has uncountable basis. The proof is based on Hecke's theory of Dirichlet series associated with modular forms for the groups $G(\lambda)$. This is in constrast with the situation in the extended Selberg class where there exist functional equations without non-trivial solutions. Presumably this holds for non-integer degrees $d$, but up to date was confirmed only for $0 \leqslant d \lt 5/3$. In the case of $d = 0$ or $d = 1$ the space of solutions belonging to the extended Selberg class, through non-trivial, is finite dimensional.

The main result of the paper is the following: for the Walsh transform of a wavelet type system on [0, l], there is an integrable function whose Fourier expansion with respect to the transformed system is divergent almost everywhere on [0, l]. This is an extension of the result by K. S. Kazarian and A. S. Sargsian [9] for the bounded Ciesielski system, i.e. the Walsh transform of the Franklin system.

We prove that each subset $A$ of $\mathbb{Z}$ which fulfills certain Sidon-type condition can be extended to a unique representation basis for $\mathbb{Z}$.

In this paper we study the notion of J-topological divisors of zero (J-d.t.z) in metrisable locally convex Jordan algebras. We give, among other things, several characterisations of these elements.

The main results of the paper are the following: the Fourier expansion of $f \in L_p$, $1 \lt p \lt \infty$, with respect to the Walsh transform of a wavelet type system converges a.e. to $f$ and if $f \in L_1$ then the same is true for the Cesàro means.

In this paper, we develop a large sieve type inequality for some special characters whare moduli are squares of primes. Our result gives a non-trivial estimate in certain ranges.