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.