AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.
Abstract
The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.
AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$
K
r
+
i
t
(
y
)
with positive, real argument y and of large complex order $$r+it$$
r
+
i
t
where r is bounded and $$t = y \sin \theta $$
t
=
y
sin
θ
for a fixed parameter $$0\le \theta \le \pi /2$$
0
≤
θ
≤
π
/
2
or $$t= y \cosh \mu $$
t
=
y
cosh
μ
for a fixed parameter $$\mu >0$$
μ
>
0
. In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$
K
r
+
i
t
(
y
)
as $$y \rightarrow \infty $$
y
→
∞
. When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$
E
0
(
j
)
(
z
,
r
+
i
t
)
for each inequivalent cusp $$\kappa _j$$
κ
j
when $$1/2 \le r \le 3/2$$
1
/
2
≤
r
≤
3
/
2
.
The coarse orbit types of non-Kählerian, symplectic homogeneous
spaces whose transformation groups are non-compact simple and isotropy
subgroups are compact