AbstractLet $$\theta $$
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.
A q -Raabe formula and an integral of the fourth Jacobi theta function
