bluuebunny/crossref_metadata_2025_split huggingface dataset:
Jacobi polynomials as generalized Faber polynomials
Authors: Zayed Ahmed I. 
Published: June 2012

Let

B

{\mathbf {B}}

be an open bounded subset of the complex

z
z

-plane with closure

B

¯

\overline {\mathbf {B}}

whose complement

B

¯

c

{\overline {\mathbf {B}} ^c}

is a simply connected domain on the Riemann sphere.

z
=
ψ
(
w
)

z = \psi (w)

map the domain

|
w
|

>
ρ

(
ρ
>
0
)

\left | w \right | > \rho \quad (\rho > 0)

one-to-one conformally onto the domain

B

¯

c

{\overline {\mathbf {B}} ^c}

such that

ψ
(
∞
)
=
∞

\psi (\infty ) = \infty

. Let

R
(
w
)
=

∑

n
=
0

∞

c
n

w

−
n

R(w) = \sum \nolimits _{n = 0}^\infty {{c_n}{w^{ - n}}}

,

c
0

≠
0

{c_0} \ne 0

be analytic in the domain

|
w
|

>
ρ

\left | w \right | > \rho

with

R
(
w
)
≠
0

R(w) \ne 0

. Let

F
(
z
)
=

∑

n
=
0

∞

b
n

z
n

F(z) = \sum \nolimits _{n = 0}^\infty {{b_n}} {z^n}

,

F
∗
(
z
)
=

∑

n
=
0

∞

1

b
n

z
n

F*(z) = \sum \nolimits _{n = 0}^\infty {\frac {1} {{{b_n}}}} {z^n}

be analytic in

|
z
|

>
1

\left | z \right | > 1

and analytically continuable to any point outside

|
z
|

>
1

\left | z \right | > 1

along any path not passing through the points

z
=
0
,
1
,
∞

z = 0,1,\infty

. The generalized Faber polynomials

{

P
n

(
z
)

}

n
=
0

∞

\{ {P_n}(z)\} _{n = 0}^\infty

of

B

{\mathbf {B}}

are defined by
\[

t

ψ
′

(
t
)

ψ
(
t
)

R
(
t
)
F

(

z

ψ
(
t
)

)

=

∑

n
=
0

∞

P
n

(
z
)

1

t
n

,

|
t
|

>
ρ

\frac {{t\psi ’(t)}} {{\psi (t)}}R(t)F\left ( {\frac {z} {{\psi (t)}}} \right ) = \sum \limits _{n = 0}^\infty {{P_n}(z)\frac {1} {{{t^n}}},} \quad \left | t \right | > \rho

\]
. The aim of this paper is to show that (1) if the Jacobi polynomials

{

P
n

(
α
,
β
)

(
z
)

}

n
=
0

∞

\{ P_n^{(\alpha ,\beta )}(z)\} _{n = 0}^\infty

are generalized Faber polynomials of any region

B

{\mathbf {B}}

, then it must be the elliptic region

{
z
:

|

z
+
1

|

+

|

z
−
1

|

>
ρ
+

1
ρ

,
ρ
>
1
}
;

\{ z:|z + 1| + |z - 1| > \rho + \frac {1}{\rho },\rho > 1\} ;

(2) the only Jacobi polynomials that can be classified as generalized Faber polynomials are the Tchebycheff polynomials of the first kind, some normalized Gegenbauer polynomials, some normalized Jacobi polynomials of type

{

P
n

(
α
,
α
+
1
)

(
z
)

}

n
=
0

∞

\{ P_n^{(\alpha ,\alpha + 1)}(z)\} _{n = 0}^\infty

,

{

P
n

(
β
+
1
,
β
)

(
z
)

}

n
=
0

∞

\{ P_n^{(\beta + 1,\beta )}(z)\} _{n = 0}^\infty

and there are no others, no matter how one normalizes them; (3) the Hermite and Laguerre polynomials cannot be generalized Faber polynomials of any region.
Iterated spinning and homology spheres
Authors: Suciu Alexander I. 
Published: June 2012

Given a closed

n
n

-manifold

M
n

{M^n}

and a tuple of positive integers

P
P

, let

σ
P

M

{\sigma _P}M

be the

P
P

-spin of

M
M

. If

M
n

⋍̸

S
n

{M^n} \not \backsimeq {S^n}

and

P
≠
Q

P \ne Q

(as unordered tuples), it is shown that

σ
P

M
⋍̸

σ
Q

M

{\sigma _P}M\not \backsimeq {\sigma _Q}M

if either (1)

H
∗

(

M
n

)
≇

H
∗

(

S
n

)

{H_*}({M^n})\not \cong {H_*}({S^n})

, (2)

π
1

M

{\pi _1}M

finite, (3)

M
M

aspherical, or (4)

n
=
3

n = 3

. Applications to the homotopy classification of homology spheres and knot exteriors are given.
Domain-independent upper bounds for eigenvalues of elliptic operators
Authors: Hook Stephen M. 
Published: June 2012

Let

Ω
⊆

R

m

\Omega \subseteq {\mathbb {R}^m}

be a bounded open set,

∂
Ω

\partial \Omega

its boundary and

Δ
\Delta

the Laplacian on

R

m

{\mathbb {R}^m}

. Consider the elliptic differential equation: (1)
\[

−
Δ
u
=
λ
u

in

Ω
;

u
=
0

on

∂
Ω
.

- \Delta u = \lambda u\quad {\text {in}}\;\Omega ;\qquad u = 0\quad {\text {on}}\;\partial \Omega .

\]
It is known that the eigenvalues,

λ
i

{\lambda _i}

, of (1) satisfy (2)
\[

∑

i
=
1

n

λ
i

λ

n
+
1

−

λ
i

⩾

m
n

4

\sum \limits _{i = 1}^n {\frac {{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac {{mn}} {4}

\]
provided that

λ

n
+
1

>

λ
n

{\lambda _{n + 1}} > {\lambda _n}

. In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.
Abelian and nondiscrete convergence groups on the circle
Authors: Hinkkanen A. 
Published: June 2012

A group

G
G

of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of

G
G

contains a subsequence, say

g

n

{{\text {g}}_n}

, such that either (i)

g

n

→

g

{{\text {g}}_n} \to {\text {g}}

and

g

n

−
1

→

g

−
1

{\text {g}}_n^{ - 1} \to {{\text {g}}^{ - 1}}

uniformly on the circle where

g

{\text {g}}

is a homeomorphism, or (ii)

g

n

→

x

0

{{\text {g}}_n} \to {{\text {x}}_0}

and

g

n

−
1

→

y

0

{\text {g}}_n^{ - 1} \to {{\text {y}}_0}

uniformly on compact subsets of the complements of

{

y

0

}

\{ {{\text {y}}_0}\}

and

{

x

0

}

\{ {{\text {x}}_0}\}

, respectively, for some points

x

0

{{\text {x}}_0}

and

y

0

{{\text {y}}_0}

of the circle (possibly

x

0

=

y

0

{{\text {x}}_0}{\text { = }}{{\text {y}}_0}

). For example, a group of

K
K

-quasisymmetric maps, for a fixed

K
K

, is a convergence group. We show that if

G
G

is an abelian or nondiscrete convergence group, then there is a homeomorphism

f
f

such that

f
∘
G
∘

f

−
1

f \circ G \circ {f^{ - 1}}

is a group of Màbius transformations.
Weighted norm estimates for the Fourier transform with a pair of weights
Authors: Strömberg Jan-Olov , Wheeden Richard L. 
Published: June 2012

We prove weighted norm inequalities of the form
\[

‖

f
^

‖

L
u
q

≤
C

‖
f
‖

H
υ
p

,

0
>
p
≤
q
>
∞
,

{\left \| {\hat f} \right \|_{L_u^q}} \leq C{\left \| f \right \|_{H_\upsilon ^p}},\quad 0 > p \leq q > \infty ,

\]
for the Fourier transform on

R

n

{{\mathbf {R}}^n}

. For some weight functions

υ
\upsilon

, the Hardy space

H
υ
p

H_\upsilon ^p

on the right can be replaced by

L
υ
p

L_\upsilon ^p

. The proof depends on making an atomic decomposition of

f
f

and using cancellation properties of the atoms.
The Arf and Sato link concordance invariants
Authors: Sturm Beiss Rachel 
Published: June 2012

The Kervaire-Arf invariant is a

Z

/

2

Z/2

valued concordance invariant of knots and proper links. The

β
\beta

invariant (or Sato’s invariant) is a

Z
Z

valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants

{

β
i

}

\{ {\beta _i}\}

associated with a two component link of linking number zero where each

β
i

{\beta _i}

is a

Z
Z

valued concordance invariant and

β
0

=
β

{\beta _0} = \beta

. In this paper we demonstrate a formula for the Arf invariant of a two component link

L
=
X
∪
Y

L = X \cup Y

of linking number zero in terms of the

β
\beta

invariant of the link:
\[

arf
⁡
(
X
∪
Y
)
=
arf
⁡
(
X
)
+
arf
⁡
(
Y
)
+
β
(
X
∪
Y
)

(
mod
2
)
.

\operatorname {arf} (X \cup Y) = \operatorname {arf} (X) + \operatorname {arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).

\]
This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link’s components. We then establish a formula for

|

β

|

|\beta |

in terms of the link’s Alexander polynomial

Δ
(
x
,
y
)
=
(
x
−
1
)
(
y
−
1
)
f
(
x
,
y
)

\Delta (x,y) = (x - 1)(y - 1)f(x,y)

:
\[

|

β
(
L
)

|

=

|

f
(
1
,
1
)

|

.

|\beta (L)| = |f(1,1)|.

\]
Finally we find a relationship between the

β
i

{\beta _i}

invariants and linking numbers of lifts of

X
X

and

Y
Y

in a

Z

/

2

Z/2

cover of the compliment of

X
∪
Y

X \cup Y

.
Upper bounds for ergodic sums of infinite measure preserving transformations
Authors: Aaronson Jon , Denker Manfred 
Published: June 2012

For certain conservative, ergodic, infinite measure preserving transformations

T
T

we identify increasing functions

A
A

, for which
\[

lim sup

n
→
∞

1

A
(
n
)

∑

k
=
1

n

f
∘

T
k

=

∫
X

f
d
μ

a

.e

.

\limsup \limits _{n \to \infty } \frac {1} {{A(n)}}\sum \limits _{k = 1}^n {f \circ } {T^k} = \int _X {fd\mu } \quad {\text {a}}{\text {.e}}{\text {.}}

\]
holds for any nonnegative integrable function

f
f

. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.
Resonance and the second BVP
Authors: Shapiro Victor L. 
Published: June 2012

Let

Ω
⊂

R

N

\Omega \subset {\mathbb {R}^N}

be a bounded open connected set with the cone property, and let

1
>
p
>
∞

1 > p > \infty

. Also, let

Q
u

Qu

be the

2
m

2m

th order quasilinear differential operator in generalized divergence form:
\[

Q
u
=

∑

1
≤

|

α

|

≤
m

(
−
1
)

|

α

|

D
α

A
α

(
x
,

ξ
m

(
u
)
)

,

Qu = \sum \limits _{1 \leq |\alpha | \leq m} {{{(- 1)}^{|\alpha |}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))},

\]
where for

u
∈

W

m
,
p

u \in {W^{m,p}}

,

ξ
m

(
u
)
=
{

D
α

u
:

|

α

|

≤
m
}

{\xi _m}(u) = \{ {D^\alpha }u:|\alpha | \leq m\}

. (For

m
=
1

m = 1

,

Q
u
=
−

∑

i
=
1

N

A
i

(
x
,
u
,
D
u
)

Qu = - \sum \nolimits _{i = 1}^N {{A_i}(x,u,Du)}

.) Under four assumptions on

A
α

{A_\alpha }

—Carathéodory, growth, monotonicity for

|

α

|

=
m

|\alpha | = m

, and ellipticity—results at resonance are established for the equation

Q
u
=
G
+
f
(
x
,
u
)

Qu = G + f(x,u)

, where

G
∈

[

W

m
,
p

(
Ω
)

]
∗

G \in {[{W^{m,p}}(\Omega )]^\ast }

and

f
(
x
,
u
)

f(x,u)

satisfies a one-sided condition (plus others). For the case

m
=
1

m = 1

, these results are tantamount to generalized solutions of the second

BVP
\text {BVP}

.
Adapted sets of measures and invariant functionals on 𝐿^{𝑝}(𝐺)
Authors: Nillsen Rodney 
Published: June 2012

Let

G
G

be a locally compact group. If

G
G

is compact, let

L
0
p

(
G
)

L_0^p(G)

denote the functions in

L
p

(
G
)

{L^p}(G)

having zero Haar integral. Let

M
1

(
G
)

{M^1}(G)

denote the probability measures on

G
G

and let

P

1

(
G
)
=

M
1

(
G
)
∩

L
1

(
G
)

{\mathcal {P}^1}(G) = {M^1}(G) \cap {L^1}(G)

. If

S
⊆

M
1

(
G
)

S \subseteq {M^1}(G)

, let

Δ
(

L
p

(
G
)
,
S
)

\Delta ({L^p}(G),S)

denote the subspace of

L
p

(
G
)

{L^p}(G)

generated by functions of the form

f
−
μ
∗
f

f - \mu \ast f

,

f
∈

L
p

(
G
)

f \in {L^p}(G)

,

μ
∈
S

\mu \in S

. If

G
G

is compact,

Δ
(

L
p

(
G
)
,
S
)
⊆

L
0
p

(
G
)

\Delta ({L^p}(G),S) \subseteq L_0^p(G)

. When

G
G

is compact, conditions are given on

S
S

which ensure that for some finite subset

F
F

of

S
S

,

Δ
(

L
p

(
G
)
,
F
)
=

L
0
p

(
G
)

\Delta ({L^p}(G),F) = L_0^p(G)

for all

1
>
p
>
∞

1 > p > \infty

. The finite subset

F
F

will then have the property that every

F
F

-invariant linear functional on

L
p

(
G
)

{L^p}(G)

is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if

1
≤
p
≤
∞

1 \leq p \leq \infty

, conditions are given upon

G
G

, and upon subsets

S
S

of

M
1

(
G
)

{M^1}(G)

whose elements satisfy certain growth conditions, which ensure that

L
p

(
G
)

{L^p}(G)

has discontinuous,

S
S

-invariant linear functionals. The results are applied to show that for

1
≤
p
≤
∞

1 \leq p \leq \infty

,

L
p

(

R

)

{L^p}(\mathbb {R})

has an infinite, independent family of discontinuous translation invariant functionals which are not

P

1

(

R

)

{\mathcal {P}^1}(\mathbb {R})

-invariant.
Polynomial solutions to constant coefficient differential equations
Authors: Smith S. Paul 
Published: June 2012

Let

D
1

,
…
,

D
r

∈

C

[
∂

/

∂

x
1

,
…
,
∂

/

∂

x
n

]

{D_1}, \ldots ,{D_r} \in \mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]

be constant coefficient differential operators with zero constant term. Let
\[

S
=
{
f
∈

C

[

x
1

,
…
,

x
n

]

|

D
j

(
f
)
=
0

for all

1
⩽
j
⩽
r
}

S = \{ f \in \mathbb {C}[{x_1}, \ldots ,{x_n}]|{D_j}(f) = 0\;{\text {for all }}1 \leqslant j \leqslant r\}

\]
be the space of polynomial solutions to the system of simultaneous differential equations

D
j

(
f
)
=
0

{D_j}(f) = 0

. It is proved that

S
S

is a module over

D

(
V
)

\mathcal {D}(V)

, the ring of differential operators on the affine scheme

V
V

with coordinate ring

C

[
∂

/

∂

x
1

,
…
,
∂

/

∂

x
n

]

/

⟨

D
1

,
…
,

D
r

⟩

\mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left \langle {{D_1}, \ldots ,{D_r}} \right \rangle

. If

V
V

is smooth and irreducible, then

S
S

is a simple

D

(
V
)

\mathcal {D}(V)

-module,

S
=
1.

D

(
V
)

S = 1.\mathcal {D}(V)

, and the generators for

D

(
V
)

\mathcal {D}(V)

yield an algorithm for obtaining a basis for

S
S

. If

V
V

is singular, then

S
S

need not be simple. However,

S
S

is still a simple

D

(
V
)

\mathcal {D}(V)

-module for certain curves

V
V

, and certain homogeneous spaces

V
V

, and this allows one to obtain a basis for

S
S

, through knowledge of

D

(
V
)

\mathcal {D}(V)

.
Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences
Authors: de la Peña J. A. , Simson D. 
Published: June 2012

Let

A
,

B

A,\;B

be artinian rings and let

A

M
B

_A{M_B}

be an

(
A
−
B
)

(A - B)

-bimodule which is a finitely generated left

A
A

-module and a finitely generated right

B
B

-module. A right

A

M
B

_A{M_B}

-prinjective module is a finitely generated module

X
R

=
(

X
A
′

,

X
B

,
φ
:

X
A
′

⊗
A

M
B

→

X
B

)

{X_R} = (X_A’, X_B, \varphi :X_A’ \otimes _A M_B \to X_B)

over the triangular matrix ring
\[

R
=

(

A

a
m
p
;

A

M
B

0

a
m
p
;
B

)

R = \left ( {\begin {array}{*{20}{c}} A & {_A{M_B}} \\ 0 & B \\ \end {array} } \right )

\]
such that

X
A
′

X_A’

is a projective

A
A

-module,

X
B

X_B

is an injective

B
B

-module, and

φ
\varphi

is a

B
B

-homomorphism. We study the category

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

of right

A

M
B

_A{M_B}

-prinjective modules. It is an additive Krull-Schmidt subcategory of

mod
(
R
)

\bmod (R)

closed under extensions. For every

X
,

Y

X,\;Y

in

prin
⁡
(
R

)
B
A

,

Ext
R
2

⁡
(
X
,

Y
)
=
0

\operatorname {prin} (R)_B^A,\;\operatorname {Ext} _R^2(X,\,Y) = 0

. When

R
R

is an Artin algebra, the category

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that

R
R

is an algebra over an algebraically closed field we give conditions for

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.
The “Defektsatz” for central simple algebras
Authors: Gräter Joachim 
Published: June 2012

LetQQbe a central simple algebra finite-dimensional over its centerFFand letVVbe a valuation ring ofFF. ThenVVhas an extension toQQ, i.e., there exists a Dubrovin valuation ringBBofQQsatisfyingV=F∩BV= F \cap B. Generally, the number of extensions ofVVtoQQis not finite and therefore the so-called intersection property of Dubrovin valuation ringsB1,…,Bn{B_1}, \ldots ,{B_n}is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersectionB1∩⋯∩Bn{B_1} \cap \cdots \; \cap {B_n}. It is shown that there exists a uniquely determined natural numbernndepending only onVVand having the following property: IfB1,…,Bk{B_1}, \ldots ,{B_k}are extensions ofVVhaving the intersection property thenk≤nk \leq nandk=nk= nholds if and only ifB1∩⋯∩Bk{B_1} \cap \cdots \cap {B_k}is integral overVV. Letnnbe the extension number ofVVtoQQ. There exist extensionsB1,⋯,Bn{B_1}, \cdots ,{B_n}ofVVhaving the intersection property and ifR1,…,Rn{R_1}, \ldots ,{R_n}are also extensions ofVVhaving the intersection property thenB1∩⋯∩Bn{B_1} \cap \cdots \cap {B_n}andR1∩⋯∩Rn{R_1} \cap \cdots \cap {R_n}are conjugate. The main result regarding the extension number is the Defektsatz:[Q:F]=fB(Q/F)eB(Q/F)n2pd[Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}, wherefB(Q/F){f_B}(Q/F)is the residue degree,eB(Q/F){e_B}(Q/F)the ramification index,nnthe extension number,p=char⁡(V/J(V))p = \operatorname {char}(V/J(V)), anddda natural number.
Convolution and hypergroup structures associated with a class of Sturm-Liouville systems
Authors: Connett William C. , Markett Clemens , Schwartz Alan L. 
Published: June 2012

Product formulas of the type
\[

u
k

(
θ
)

u
k

(
ϕ
)
=

∫
0
π

u
k

(
ξ
)
D
(

ξ
,
θ
,
ϕ
)

d
ξ

{u_k}(\theta ){u_k}(\phi ) = \int _0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi

\]
are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on

[
0
,
π
]

[0,\pi ]

by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When

D
(
ξ
,
θ
,
ϕ
)

D(\xi ,\theta ,\phi )

is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which

M
[
0
,
π
]

M[0,\pi ]

becomes a Banach algebra with the functions

u
k

(
ξ
)

/

u
0

(
ξ
)

{u_k}(\xi )/{u_0}(\xi )

as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.
On the Θ-function of a Riemannian manifold with boundary
Authors: Hsu Pei 
Published: June 2012

Let

Ω
\Omega

be a compact Riemannian manifold of dimension

n
n

with smooth boundary. Let

λ
1

>

λ
2

≤
⋯

{\lambda _1} > {\lambda _2} \leq \cdots

be the eigenvalues of the Laplace-Beltrami operator with the boundary condition

[
∂

/

∂
n
+
γ
]
ϕ
=
0

[\partial /\partial n + \gamma ]\phi = 0

. The associated

Θ
\Theta

-function

Θ
γ

(
t
)
=

∑

n
=
1

∞

exp
⁡
[
−

λ
n

t
]

{\Theta _\gamma }(t) = \sum \nolimits _{n = 1}^\infty {\exp [ - {\lambda _n}t]}

has an asymptotic expansion of the form
\[

(
4
π
t

)

n

/

2

Θ
γ

(
t
)
=

a
0

+

a
1

t

1

/

2

+

a
2

t
+

a
3

t

3

/

2

+

a
4

t
2

+
⋯
.

{(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .

\]
The values of

a
0

{a_0}

,

a
1

{a_1}

are well known. We compute the coefficients

a
2

{a_2}

and

a
3

{a_3}

in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel

p
(
t
,
x
,
y
)

p(t,x,y)

near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].
Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
Authors: Runckel Hans-J. 
Published: June 2012

For the limit periodic

J
J

-fraction

K
(
−

a
n

/

(
λ
+

b
n

)
)

K( - {a_n}/(\lambda + {b_n}))

,

a
n

{a_n}

,

b
n

∈

C

{b_n} \in \mathbb {C}

,

n
∈

N

n \in \mathbb {N}

, which is normalized such that it converges and represents a meromorphic function

f
(
λ
)

f(\lambda )

on

C

∗

:=

C

∖
[
−
1
,
1
]

{\mathbb {C}^{\ast } }: = \mathbb {C}\backslash [ - 1,1]

, the numerators

A
n

{A_n}

and denominators

B
n

{B_n}

of its

n
n

th approximant are explicitly determined for all

n
∈

N

n \in \mathbb {N}

. Under natural conditions on the speed of convergence of

a
n

{a_n}

,

b
n

{b_n}

,

n
→
∞

n \to \infty

, the asymptotic behaviour of the orthogonal polynomials

B
n

{B_n}

,

A

n
+
1

{A_{n + 1}}

(of first and second kind) is investigated on

C

∗

{\mathbb {C}^{\ast } }

and

[
−
1
,
1
]

[ - 1,1]

. An explicit representation for

f
(
λ
)

f(\lambda )

yields continuous extension of

f
f

from

C

∗

{\mathbb {C}^{\ast } }

onto upper and lower boundary of the cut

(
−
1
,
1
)

( - 1,1)

. Using this and a determinant relation, which asymptotically connects both sequences

A
n

{A_n}

,

B
n

{B_n}

, one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences

B
n

{B_n}

,

A

n
+
1

{A_{n + 1}}

,

n
∈

N

n \in \mathbb {N}

. This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for

f
(
λ
)

f(\lambda )

yields meromorphic extension of

f
f

from

C

∗

{\mathbb {C}^{\ast } }

across

(
−
1
,
1
)

( - 1,1)

onto a region of a second copy of

C

\mathbb {C}

which there is bounded by an ellipse, whose focal points

±
1

\pm 1

are first order algebraic branch points for

f
f

. Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions

K
(
−

a
n

(
z
)

/

(
λ
(
z
)
+

b
n

(
z
)
)
)

K( - {a_n}(z)/(\lambda (z) + {b_n}(z)))

, where

a
n

(
z
)

{a_n}(z)

,

b
n

(
z
)

{b_n}(z)

,

λ
(
z
)

\lambda (z)

are holomorphic on a region in

C

\mathbb {C}

. Finally, for

T
T

-fractions

T
(
z
)
=
K
(
−

c
n

z

/

(
1
+

d
n

z
)
)

T(z) = K( - {c_n}z/(1 + {d_n}z))

with

c
n

→
c

{c_n} \to c

,

d
n

→
d

{d_n} \to d

,

n
→
∞

n \to \infty

, the exact convergence regions are determined for all

c
c

,

d
∈

C

d \in \mathbb {C}

. Again, explicit representations for

T
(
z
)

T(z)

yield continuous and meromorphic extension results. For all

c
c

,

d
∈

C

d \in \mathbb {C}

the regions (on Riemann surfaces) onto which

T
(
z
)

T(z)

can be extended meromorphically, are described explicitly.
Spectral multiplicity for 𝐺𝑙_{𝑛}(𝑅)
Authors: Huntley Jonathan 
Published: June 2012

We study the behavior of the cuspidal spectrum of

Γ
∖

H

\Gamma \backslash \mathcal {H}

, where

H

\mathcal {H}

is associated to

Gl
n

⁡
(
R
)

\operatorname {Gl}_n(R)

and

Γ
\Gamma

is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if

r
r

is the dimension of

H

\mathcal {H}

,

N
α

(
λ
)

{N_\alpha }(\lambda )

is the counting function for the Laplacian attached to a Hilbert space

H
α

{H_\alpha }

,

M
α

(
λ
)

{M_\alpha }(\lambda )

is the multiplicity function, and

H
0

{H_0}

is the space of cusp forms, we obtain the following results: Theorem 1. There exists a space of functions

H
1

{H^1}

, containing all cusp forms, such that
\[

N
′
(
λ
)
=

C
r

(

Vol

X
)

λ

r
2

+
O
(

λ

r
−
1

2

λ

1

n
+
1

(
log
⁡
λ

)

n
−
1

)
.

N\prime (\lambda ) = {C_r}({\text {Vol}}\;X){\lambda ^{\frac {r} {2}}} + O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).

\]
Theorem 2.
\[

M
0

(
λ
)
=
O
(

λ

r
−
1

2

λ

1

n
+
1

(
log
⁡
λ

)

n
−
1

)
.

{M_0}(\lambda ) = O({\lambda ^{\frac {{r - 1}} {2}}}{\lambda ^{\frac {1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).

\]