The classical Fej\'er's theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.

We consider$Thurston maps$, i.e., branched covering maps$f:S$^{2}→$S$^{2}that are$post-critically finite$. In addition, we assume that$f$is$expanding$in a suitable sense. It is shown that each sufficiently high iterate$F$=$f$^{$n$}of$f$is$semi-conjugate$to$z$^{$d$}:$S$^{1}→$S$^{1}, where$d$= deg F. More precisely, for such an$F$we construct a$Peano curve γ$:$S$^{1}→$S$^{2}(onto), such that$F$∘$γ$($z$) =$γ$($z$^{$d$}) (for all$z$∈$S$^{1}).

Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.