The spectral decomposition of $$|\theta |^2$$

Let $$\theta $$ θ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $$\langle |\theta |^2, \varphi \rangle $$ ⟨ | θ | 2 , φ ⟩ as $$\varphi $$ φ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $$|\theta |^2 \notin L^2$$ | θ | 2 ∉ L 2 . Applications to the subconvexity, quantum variance and 4-norm problems We determine all pairs $$(A_{f,g},A_{g,h})$$ ( A f , g , A g , h ) of generalized weighted quasi-arithmetic means being square iterative roots of $$(A_{F,G},A_{G,H})$$ ( A F , G , A G , H ) , that is, the equation $$( A_{f,g},A_{g,h}) \circ ( A_{f,g},A_{g,h}) =(A_{F,G},A_{G,H}),$$ ( A f , g , A g , h ) ∘ ( A f , g , A g , h ) = ( A F , G , A G , H ) , is solved under three times differentiability of the functions f, g, h, F, G, H. As an application, some special cases are presented. are indicated.

Subcommuting and comparable iterative roots of order preserving homeomorphisms

It is known that the iterative roots of continuous functions are not necessarily unique, if it exist. In this note, by introducing the set of points of coincidence, we study the iterative roots of order preserving homeomorphisms. In particular, we prove a characterization of identical iterative roots of an order preserving homeomorphism using the points of coincidence of functions.