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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  for the bounded Ciesielski system, i.e. the Walsh transform of the Franklin system.
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.