bluuebunny/crossref_metadata_2025_split huggingface dataset:
Minimal functions, martingales, and Brownian motion on a noncompact symmetric space
Authors: Taylor J. C. 
Published: June 2010

A Brownian motion on

R

n

{{\mathbf {R}}^n}

may be characterized as a process

(

X
t

)

t
⩾
0

{({X_t})_{t \geqslant 0}}

on a probability space

(
Ω
,

F

,
P
)

(\Omega ,\mathfrak {F},P)

such that, for all

y
∈

R

d

,
exp
⁡

{

−
(
t

/

2
)

|

|

y

|

|

2

+

⟨

y
,

X
t

⟩

}

y \in {{\mathbf {R}}^d},\exp \left \{ { - (t/2)||y|{|^2} + \left \langle {y,{X_t}} \right \rangle } \right \}

is a martingale of expectation one. The analogue of this fact is proved for the Brownian motion on a noncompact symmetric space.
Regularizing properties of nonlinear semigroups
Authors: Gutman Semion 
Published: June 2010

It is known that some classes of

m
m

-accretive operators

A
A

generate Lipschitz continuous semigroups of contractions; that is

|

|

S
(
t
+
h
)
x
−
S
(
t
)
x

|

|

⩽
L
(

|

|

x

|

|

)
h

/

t
,
0
⩽
t
⩽
t
+
h
⩽
T
,
x
∈

D
(
A
)

¯

||S(t + h)x - S(t)x|| \leqslant L(||x||)h/t,0 \leqslant t \leqslant t + h \leqslant T,x \in \overline {D(A)}

. If the underlying Banach spaces

X
X

and

X
∗

{X^*}

are uniformly convex and an

m
m

-accretive operator

B
B

is bounded, we prove, in particular, that the semigroup generated by

A
+
B

A + B

is Hölder continuous. The proof is based on a result on the structure of accretive operators obtained via the Kuratowski-Ryll-Nardzewski Selection Theorem. Also, we consider some applications of these results to the existence of solutions of

u
′

+
A
u
+
B
u
∍
C
u
,
u
(
0
)
=

u
0

u’ + Au + Bu \backepsilon Cu,u(0) = {u_0}

.
Cyclic vectors in 𝐴^{-∞}
Authors: Brown Leon , Korenblum Boris 
Published: June 2010

If

f
f

is in

A

−
p

{A^{ - p}}

, then

f
f

is cyclic in

A

−
∞

{A^{ - \infty }}

if and only if

f
f

is cyclic in every

A

−
q

(
q

>

p
)

{A^{ - q}}(q{\text { > }}p)

. An analogous result holds for the Bergman spaces

B
p

{B^p}

. In this note we apply the theory developed in [2 and 3] to explain the relationship between cyclic vectors in

A

−
∞

{A^{ - \infty }}

and

A

−
p

{A^{ - p}}

or

B
p

{B^p}

.
On the sphere conjecture of Birkhoff
Authors: Jerrard Richard 
Published: June 2010

Birkhoff’s sphere conjecture, now known to be false, says that if

f
f

is a measure preserving homeomorphism of

S
2

{S^2}

with the poles

N
N

and

S
S

fixed, and with no other periodic points, then

f
f

is topologically conjugate to an irrational rotation of

S
2

{S^2}

. In this setting we say that

D
⊂

S
2

D \subset {S^2}

is maximal for

f
f

if

f

(
D
)

∩
D
=
∅

f\left ( D \right ) \cap D = \emptyset

and

D
D

is maximal with respect to that property. Also,

f
f

is

2
2

-small if for any circular ball

B
B

such that

f

(
B
)

∩
B
=
∅
,

f

−
1

(
B
)

∩
f

(
B
)

=
∅

f\left ( B \right ) \cap B = \emptyset ,{f^{ - 1}}\left ( B \right ) \cap f\left ( B \right ) = \emptyset

also. Theorem. Any

f
f

as above and also

2
2

-small has a maximal set

D
D

which is an open ball; its boundary contains

N
N

and

S
S

and is locally connected, and the area of

D
D

is an irrational fraction of the area of

S
2

{S^2}

. This theorem gives another way of looking at the maps involved in the Birkhoff conjecture.
Note on multiplicity
Authors: Katz Daniel 
Published: June 2010

Let

(
R
,
M
)

(R,M)

be a local ring with infinite residue field and

I
=
(

x
1

,
…
,

x
d

)
R

I = ({x_1}, \ldots ,{x_d})R

an ideal generated by a system of parameters. It is shown that the multiplicity of

I
I

equals the multiplicity of

I
T

IT

where
\[

T
=

R
~

[

x
1

/

x
d

,
…
,

x

d
−
1

/

x
d

]

M

R
~

[

x
1

/

x
d

,
…
,

x

d
−
1

/

x
d

]

T = \tilde R{[{x_1}/{x_d}, \ldots ,{x_{d - 1}}/{x_d}]_{M\tilde R[{x_1}/{x_d}, \ldots ,{x_{d - 1}}/{x_d}]}}

\]
and

R
~

=
R

/

(
0
:

x
d
N

)
,
N

\tilde R = R/(0:x_d^N),N

large.
Growth properties of 𝑝th means of potentials in the unit ball
Authors: Gardiner S. J. 
Published: June 2010

Let

v
v

be a potential in the unit ball of

R

n

{{\mathbf {R}}^n}

, and

M

p

(
v
;
r
)

{\mathcal {M}_p}(v;r)

be its

p
p

th order mean over the sphere of radius

r
r

centred at the origin. It is shown that, as

r
→
1
−

r \to 1 -

, the function

f
(
r
)
=

(
1
−
r

)

(
n
−
1
)
(
1
−
1

/

p
)

M

p

(
v
;
r
)

f(r) = {(1 - r)^{(n - 1)(1 - 1/p)}}{\mathcal {M}_p}(v;r)

has limit 0 when

1
≤
p

>

(
n
−
1
)

/

(
n
−
2
)

1 \leq p{\text { > }}(n - 1)/(n - 2)

, and has lower limit 0 when

n
≥
3

n \geq 3

and

(
n
−
1
)

/

(
n
−
2
)
≤
p

>

(
n
−
1
)

/

(
n
−
3
)

(n - 1)/(n - 2) \leq p{\text { > }}(n - 1)/(n - 3)

. This extends a result of Stoll, who showed that, when

n
=
2

n = 2

and

p
=
+
∞
,
lim

inf

r
→
1
−

f
(
r
)
=
0

p = + \infty ,\lim {\inf _{r \to 1 - }}f(r) = 0

. Examples are given to show that the theorems presented are best possible.
Differentiability of distance functions and a proximinal property inducing convexity
Authors: Giles J. R. 
Published: June 2010

In a normed linear space

X
X

, consider a nonempty closed set

K
K

which has the property that for some

r
>
0

r > 0

there exists a set of points

x
0

∈
X
∖
K
,
d
(

x
0

K
)
>
r

{x_0} \in X\backslash K,d({x_0}K) > r

, which have closest points

p
(

x
0

)
∈
K

p({x_0}) \in K

and where the set of points

x
0

−
r
(
(

x
0

−
p
(

x
0

)
)

/

|

|

x
0

−
p
(

x
0

)

|

|

)

{x_0} - r(({x_0} - p({x_0}))/||{x_0} - p({x_0})||)

is dense in

X
∖
K

X\backslash K

. If the norm has sufficiently strong differentiability properties, then the distance function

d
d

generated by

K
K

has similar differentiability properties and it follows that, in some spaces,

K
K

is convex.
The Wiener lemma and cocycles
Authors: Rychlik Marek Ryszard 
Published: June 2010

We give a sufficient and necessary condition for a function with its values in the unit circle to be a multiplicative coboundary. This theorem generalizes the following result of Veech [1]. Let

T
:

T

→

T

T:{\mathbf {T}} \to {\mathbf {T}}

be a rotation of the unit circle

T

{\mathbf {T}}

by an irrational angle

θ
\theta

. Let

F
:

T

→

T

F:{\mathbf {T}} \to {\mathbf {T}}

be a measurable function. Then

F
F

is a multiplicative coboundary iff
\[

∫

T

F
(
x
)
F
(
T
x
)
⋯
F
(

T

n
−
1

x
)
d
μ
(
x
)
→
1
,

as

‖

n
θ

‖

→
0
,

\int _{\mathbf {T}} {F(x)F(Tx) \cdots F({T^{n - 1}}x)d\mu (x) \to 1,\quad {\text {as }}\left \| {n\theta } \right \| \to 0,}

\]
where

‖

n
θ

‖

\left \| {n\theta } \right \|

is the distance of

n
θ

n\theta

from integers and

μ
\mu

is the Haar measure.
A semigroup treatment of a one-dimensional nonlinear parabolic equation
Authors: Serizawa Hisamitsu 
Published: June 2010

In this paper we study the existence, uniqueness and differentiability of solutions (in a certain generalized sense) for the nonlinear parabolic equation
\[

u
t

=

u

x
x

−
F
(
u
,

u
x

)
(
0
>
x
>
1
,
t
>
0
)
,

{u_t} = {u_{xx}} - F(u,{u_x})(0 > x > 1,t > 0),

\]
under the maximal monotone boundary conditions:
\[

(
−
1

)
i

u
x

(
t
,
i
)
∈

β
i

(
u
(
t
,
i
)
)
,
t
>
0
,
i
=
0
,
1.

{( - 1)^i}{u_x}(t,i) \in {\beta _i}(u(t,i)),t > 0,i = 0,1.

\]
Group actions and direct sum decompositions of 𝐿^{𝑝} spaces
Authors: Nillsen Rodney 
Published: June 2010

Let

G
G

be a locally compact group of measure preserving transformations on a

σ
\sigma

-finite measure space

(

X
,

B

,
m

)

\left ( {X,\mathcal {B},m} \right )

, and let

S
S

be a subset of

M
1

(
G
)

{M^1}\left ( G \right )

. Let

1
>
p
>
∞
,

I
p

=

{

f
:
f
∈

L
p

(
m
)

and

g

f
=
f
,

for all

g
∈
G

}

1 > p > \infty ,{I_p} = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and}}{{\text { }}_g}f = f,{\text {for all }}g \in G} \right \}

, let

I
p

(
S
)

=

{

f
:
f
∈

L
p

(
m
)

and

μ
∗
f
=
f

for all

μ
∈
S

}

{I_p}\left ( S \right ) = \left \{ {f:f \in {L^p}\left ( m \right ){\text { and }}\mu * f = f{\text { for all }}\mu \in S} \right \}

, and let

K
p

(
S
)

{K_p}\left ( S \right )

be the closed subspace of

L
p

(
m
)

{L^p}\left ( m \right )

generated by functions of the form

μ
∗
f
−
f

\mu * f - f

, for

f
∈

L
p

(
m
)

f \in {L^p}\left ( m \right )

and

μ
∈
S

\mu \in S

. Conditions are given on

S
S

which ensure that

I
p

=

I
p

(
S
)

{I_p} = {I_p}\left ( S \right )

, and this is used to express

L
p

(
m
)

{L^p}\left ( m \right )

as a direct sum of

I
p

{I_p}

and

K
p

(
S
)

{K_p}\left ( S \right )

.
On operators that are almost isometric on the positive cones of 𝐿^{𝑝}-spaces, 1<𝑝<+∞
Authors: Huang Sen Zhong 
Published: June 2010

A linear operator

T
:

L
p

(
Ω
)

→

L
p

(

Ω
1

)

T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )

is said to be almost isometric on the positive cone (a.i.p.c.) with distortion coefficient

ε
\varepsilon

if there is an

ε
,
0
≤
ε
≤
1

\varepsilon ,0 \leq \varepsilon \leq 1

, such that

(

1
−
ε

)

‖
f
‖

≤

‖

T
f

‖

≤

‖
f
‖

\left ( {1 - \varepsilon } \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq \left \| f \right \|

for all nonnegative functions

f
f

. We prove that if

p
∈
[
2
,
∞
)

p \in [2,\infty )

, then there are continuous functions

a
p

(
⋅
)

{a_p}\left ( \cdot \right )

and

b
p

(
⋅
)

{b_p}\left ( \cdot \right )

defined on

[
0
,
1
)

[0,1)

, with

a
p

(
0
)

=

b
p

(
0
)

=
1

{a_p}\left ( 0 \right ) = {b_p}\left ( 0 \right ) = 1

, so that if

T
:

L
p

(
Ω
)

→

L
p

(

Ω
1

)

T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )

is an a.i.p.c. operator with distortion coefficient

ε
\varepsilon

, then

a
p

(
ε
)

‖
f
‖

≤

‖

T
f

‖

≤

b
p

(
ε
)

‖
f
‖

{a_p}\left ( \varepsilon \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq {b_p}\left ( \varepsilon \right )\left \| f \right \|

for all

f
∈

L
p

(
Ω
)

f \in {L^p}\left ( \Omega \right )

. We also prove that such functions

a
p

(
⋅
)

{a_p}\left ( \cdot \right )

and

b
p

(
⋅
)

{b_p}\left ( \cdot \right )

exist for

p
p

in the range

(

1
,
2

)

\left ( {1,2} \right )

if, in addition,

T
T

is either positive or has dense range.
Uniform bounds for isoperimetric problems
Authors: Siegel Jerrold , Williams Frank 
Published: June 2010

In this paper we generalize our previous joint work with Allan Calder on the width of homotopies by considering an arbitrary finite polyhedral pair

(

W
,
V

)

\left ( {W,V} \right )

rather than

(

I
,

{

0
,
1

}

)

\left ( {I,\left \{ {0,1} \right \}} \right )

. We show that given appropriate topological conditions on a Riemannian manifold

M
M

, with respect to

(

W
.
V

)

\left ( {W.V} \right )

, there are bounds,

B
q

(

a
,

(

W
,
V

)

,
M

)

{B_q}\left ( {a,\left ( {W,V} \right ),M} \right )

, such that if

F
:
K
×
W
→
M

F:K \times W \to M

is a map with

Lip

(

F

|

(

K
×
V

)

)

>
a

{\text {Lip}}\left ( {F\left | {\left ( {K \times V} \right )} \right .} \right ) > a

, then

F
F

can be deformed

rel

(

K
×
V

)

{\text {rel}}\left ( {K \times V} \right )

to

F
′

F’

with

Lip

(

F
′

)

>

B
q

(

a
,

(

W
,
V

)

,
M

)

+
ε

{\text {Lip}}\left ( {F’} \right ) > {B_q}\left ( {a,\left ( {W,V} \right ),M} \right ) + \varepsilon

, where

ε
>
0

\varepsilon > 0

is arbitrary and

dim
⁡

(
K
)

=
q

\dim \left ( K \right ) = q

.
Remark on Walter’s inequality for Schur multipliers
Authors: Bożejko Marek 
Published: June 2010

We extend and give another proof of Walter’s inequality: For a linear operator

T
∈
L
(

l
2

(
X
)
)

T \in L({l^2}(X))

\[

|

|

T

|

|

V
2

(
X
)

2

≤

|

|

|

T

|

l

|

|

V
2

(
X
)

|

|

|

T

|

r

|

|

V
2

(
X
)

,

||T||_{{V_2}(X)}^2 \leq ||\,|T{|_l}|{|_{{V_2}(X)}}||\,|T{|_r}|{|_{{V_2}(X)}},

\]
where

V
2

(
X
)

{V_2}(X)

is the Banach algebra of Schur multipliers on

L
(

l
2

(
X
)
)

L({l^2}(X))

and

|

T

|

l

=

(
T
T

)

∗
1

/

2

,

|

T

|

r

=

|

T
∗

|

l

|T{|_l} = {(TT)^{*1/2}},|T{|_r} = |{T^*}{|_l}

.
A result about the Hilbert transform along curves
Authors: Saal Linda V. 
Published: June 2010

Let

G
G

be a connected and simply connected, nilpotent Lie group and let

γ
:
(
−
1
,
1
)
→
G

\gamma :( - 1,1) \to G

be a (connected) analytic curve such that

γ
(
0
)
=
0

\gamma (0) = 0

. Then the Hilbert transform along

γ
\gamma

,
\[

T
f
(
x
)
=
p
.
v
.

∫

0
>

|

t

|

>
1

f
(
x
γ

(
t
)

−
1

)
d
t

/

t

,

Tf(x) = p.v.\int _{0 > |t| > 1} {f(x\gamma {{(t)}^{ - 1}})dt/t} ,

\]
is bounded on

L
p

(
G
)
,
1
>
p
>
∞

{L^p}(G),1 > p > \infty

.
The Bloch constant of bounded analytic functions on a multiply connected domain
Authors: Colonna Flavia 
Published: June 2010

Let

F
F

be an analytic function on the bounded domain

R
R

, a multiply-connected region of the complex plane. The Bloch constant of

F
F

is defined by
\[

β
F

=

sup

|

z

|

>
1

(
1
−

|

z

|

2

)

|

(
F
∘
p

)
′

(
z
)

|

,

\beta _F = \sup _{|z| > 1} (1 - |z|^2)|(F \circ p)’(z)|,

\]
where

p
p

is a conformal universal cover of

R
R

with domain

Δ
\Delta

, the open unit disk. If

F
F

is bounded, then

β
F

≤

|

|

F

|

|

∞

{\beta _F} \leq ||F|{|_\infty }

, the sup-norm of

F
F

. In this paper we characterize those functions

F
F

for which

β
F

=

|

|

F

|

|

∞

{\beta _F} = ||F|{|_\infty }

in terms of the zeros of

F
F

when the boundary of

R
R

is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.
Pseudospectral operators and the pointwise ergodic theorem
Authors: Bradley R. E. 
Published: June 2010

We show that for a class of operators which properly contains the normal operators on

L
2

{L_2}

,
\[

1
n

∑

i
=
0

n
−
1

T
i

f
→
a
.
e
.

iff

1

2
n

∑

i
=
0

2
n

−
1

T
i

f
→
a
.
e
.

\frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.}

\]
This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.
Asymptotic periodicity of solutions to a class of neutral functional-differential equations
Authors: Wu Jian Hong 
Published: June 2010

In this paper, we extend a convergence result due to Takáč to continuous maps satisfying certain monotonicity properties. Applying this extension to the Poincaré map associated with the neutral equation
\[

(
d

/

d
t
)
[
x
(
t
)
−
b
(
t
)
x
(
t
−
r
)
]
=
F
[
t
,
x
(
t
)
,
x
(
t
−
r
)
]

(d/dt)[x(t) - b(t)x(t - r)] = F[t,x(t),x(t - r)]

\]
we prove that each solution of the above neutral equation tends to an

r
r

-periodic function as

t
→
∞

t \to \infty

in an oscillatory manner, where

0
≤
b
(
t
)
>
1

0 \leq b(t) > 1

is an

r
r

-periodic continuous function and

F
F

satisfies a certain order relation.
Extremal compressions of closed operators
Authors: Förster K.-H. , Jahn K. 
Published: June 2010

Let

X
X

be a Banach space,

A
A

a closed linear operator on

X
X

, and

λ
1

,
…
,

λ
n

{\lambda _1}, \ldots ,{\lambda _n}

isolated eigenvalues of

A
A

of finite multiplicity. If

P
P

is a projection on

X
X

such that

λ
1

,
…
,

λ
n

{\lambda _1}, \ldots ,{\lambda _n}

belong to the resolvent of the compression of

A
A

on the range of

P
P

it is easy to see that
\[

dim
⁡
N

(
P
)

≥
max

{

dim
⁡
N

(

λ
i

−
A

)

:
1
≤
i
≤
n

}

.

\dim N\left ( P \right ) \geq \max \left \{ {\dim N\left ( {{\lambda _i} - A} \right ):1 \leq i \leq n} \right \}.

\]
It is shown that there exist such projections where we have equality in this inequality.
Convergence and integrability of trigonometric series with coefficients of bounded variation of order (𝑚,𝑝)
Authors: Stanojevic Vera B. 
Published: June 2010

Let

{
c
(
n
)
}

\{ c(n)\}

be a complex null sequence such that for some integer

m
≥
1

m \geq 1

and some

p
∈
(
1
,
2
]

p \in (1,2]

\[

∑

|

n

|

>
∞

|

Δ
m

c
(
n
)

|

p

>
∞

and

∑

n
=
1

∞

|

Δ
(
c
(
n
)
−
c
(
−
n
)
)

|

lg
⁡
n
>
∞
.

\sum \limits _{|n| > \infty } {|{\Delta ^m}c(n){|^p} > \infty \quad {\text {and}}\quad \sum \limits _{n = 1}^\infty {|\Delta (c(n) - c( - n))|\lg n > \infty .} }

\]
It is shown that the series
\[

(
∗
)

∑

|

n

|

>
∞

c
(
n
)

e

int

,

t
∈
T
=

R

2
π

Z

( * )\quad \sum \limits _{|n| > \infty } {c(n)} {e^{\operatorname {int} }},\quad t \in T = \frac {\mathbb {R}}{{2\pi \mathbb {Z}}}

\]
converges a.e. and that the well-known condition

C
w

{C_w}

of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that

n
Δ
c
(
n
)
=
0
(
1
)
,

|

n

|

→
∞

n\Delta c(n) = 0(1),\quad |n| \to \infty

, implies that the condition

C
w

{C_w}

is equivalent to the de la Vallee Poussin summability of partial sums

{

S
n

(
c
)
}

\{ {S_n}(c)\}

as conjectured in [8].
The Hilbert transform with exponential weights
Authors: Colzani Leonardo , Vignati Marco 
Published: June 2010

We study the operator
\[

H

f
(
x
)
=

2

−
x

∫
0

+
∞

2
y

f
(
y
)

x
−
y

d
y

\mathcal {H}f(x) = {2^{ - x}}\int _0^{ + \infty } {\frac {{{2^y}f(y)}}{{x - y}}dy}

\]
on Lorentz spaces on

R

+

{\mathbb {R}_ + }

with respect to the measure

4
x

d
x

{4^x}dx

. This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this operator is bounded on the Lorentz spaces

L

2
,
9

(

R

+

,

4
x

d
x
)
,
1
>
q
>
+
∞

{L^{2,9}}({\mathbb {R}_ + },{4^x}dx),1 > q > + \infty

, and it maps the Lorentz space

L

2
,
1

(

R

+

,

4
x

d
x
)

{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)

into a space that we call WEAK-

L

2
,
1

(

R

+

,

4
x

d
x
)

{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)

. We also prove that

H

\mathcal {H}

maps

L
1

(

R

+

,

4
x

d
x
)

{L^1}({\mathbb {R}_ + },{4^x}dx)

into WEAK-

L
1

(

R

+

,

4
x

d
x
)
+

L
2

(

R

+

,

4
x

d
x
)

{L^1}({\mathbb {R}_ + },{4^x}dx) + {L^2}({\mathbb {R}_ + },{4^x}dx)

.
A sharp estimate in an operator inequality
Authors: McEachin R. 
Published: June 2010

Let

H

\mathcal {H}

and

K

\mathcal {K}

be Hilbert spaces, and suppose

A
∈

B

(

H

)

A \in \mathcal {B}(\mathcal {H})

and

B
∈

B

(

K

)

B \in \mathcal {B}(\mathcal {K})

are selfadjoint operators with

dist
⁡
(
σ
(
A
)
,
σ
(
B
)
)
≥
δ
>
0

\operatorname {dist} (\sigma (A),\sigma (B)) \geq \delta > 0

. It is known that for any

Q
∈

B

(

K

,

H

)

Q \in \mathcal {B}(\mathcal {K},\mathcal {H})

we must have

π
2

|

|

A
Q
−
Q
B

|

|

≥
δ

|

|

Q

|

|

\tfrac {\pi }{2}||AQ - QB|| \geq \delta ||Q||

. In this paper we give examples proving that

π

/

2

\pi /2

is sharp in this inequality.
Asymptotic formulas for ultraspherical polynomials 𝑃_{𝑛}^{𝜆}(𝑥) and their zeros for large values of 𝜆
Authors: Elbert Árpád , Laforgia Andrea 
Published: June 2010

For

λ
>
−
1

/

2

\lambda > - 1/2

we denote by

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

the ultraspherical polynomial of degree

n
n

and by

x

n
,
k

(
λ
)

x_{n,k}^{(\lambda )}

and

h

n
,
k

(
k
=
1
,
2
,
…
,
n
)

{h_{n,k}}(k = 1,2, \ldots ,n)

the

k
k

th zeros of

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

and of the Hermite polynomial

H
n

(
x
)

{H_n}(x)

, respectively. In this paper we establish the following formulas
\[

λ

−
n

/

2

P
n

(
λ
)

(

x

λ

)

=

∑

j
=
0

n
−
1

λ

−
j

Q

n
j

(
x
)

for

λ
≠
0

{\lambda ^{ - n/2}}P_n^{(\lambda )}\left ( {\frac {x}{{\sqrt \lambda }}} \right ) = \sum \limits _{j = 0}^{n - 1} {{\lambda ^{ - j}}{Q_{nj}}(x)\,{\text {for}}\,\lambda \ne 0}

\]
and
\[

x

n
,
k

(
λ
)

=

h

n
,
k

λ

−
1

/

2

−

h

n
,
k

8

(
2
n
−
1
+
2

h

n
,
k

2

)

λ

−
3

/

2

+

h

n
,
k

(

12

n
2

−
12
n
+
1

128

+

5
n
−
2

24

h

n
,
k

2

+

5

96

h

n
,
k

4

)

λ

−
5

/

2

+
O
(

λ

−
7

/

2

)
,
λ
→
∞

x_{n,k}^{(\lambda )} = {h_{n,k}}{\lambda ^{ - 1/2}} - \frac {{{h_{n,k}}}}{8}(2n - 1 + 2h_{n,k}^2){\lambda ^{ - 3/2}} + {h_{n,k}}\left ( {\frac {{12{n^2} - 12n + 1}}{{128}} + \frac {{5n - 2}}{{24}}h_{n,k}^2 + \frac {5}{{96}}h_{n,k}^4} \right ){\lambda ^{ - 5/2}} + O({\lambda ^{ - 7/2}}),\lambda \to \infty

\]
where

Q

n
0

(
x
)
=

H
n

(
x
)

/

n
!

{Q_{n0}}(x) = {H_n}(x)/n!

and

Q

n
j

(
x
)
(
j
=
1
,
2
,
…
,
n
−
1
)

{Q_{nj}}(x)(j = 1,2, \ldots ,n - 1)

are polynomials specified in Theorem 1. Finally we show that the positive (negative) zeros of

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

are convex (concave) functions of

λ
\lambda

, provided

λ
\lambda

is sufficiently large.
Sample path-valued conditional Yeh-Wiener integrals and a Wiener integral equation
Authors: Park Chull , Skoug David 
Published: June 2010

In this paper we evaluate the conditional Yeh-Wiener integral

E
(
F
(
x
)

|

x
(
s
,
t
)
=
ξ
)

E(F(x)|x(s,t) = \xi )

for functions

F
F

of the form
\[

F
(
x
)
=
exp
⁡
{

∫
0
t

∫
0
s

ϕ

(
σ
,
τ
,
x
(
σ
,
τ
)
)
d
σ
d
τ
}
.

F(x) = \exp \{ \int _0^t {\int _0^s \phi } (\sigma ,\tau ,x(\sigma ,\tau ))d\sigma d\tau \} .

\]
The method we use to evaluate this conditional integral is to first define a sample path-valued conditional Yeh-Wiener integral of the type

E
(
F
(
x
)

|

x
(
s
,
)
=
ψ
(
)
)

E(F(x)|x(s,) = \psi ())

and show that it satisfies a Wiener integral equation. We next obtain a series solution for

E
(
F
(
x
)

|

x
(
s
,
)
=
ψ
(
)
)

E(F(x)|x(s,) = \psi ())

by solving this Wiener integral equation. Finally, we integrate this series solution appropriately in order to evaluate

E
(
F
(
x
)

|

x
(
s
,
t
)
=
ξ
)

E(F(x)|x(s,t) = \xi )

.
One radius theorem for the eigenfunctions of the invariant Laplacian
Authors: Kwon E. G. 
Published: June 2010

Let

B
B

be the open unit ball in

C

n

{{\mathbf {C}}^n}

with its boundary

S
S

. Suppose that

α
≥

1
2

\alpha \geq \tfrac {1}{2}

and

u
(
z
)
=

(
1
−

|
z
|

2

)

n
(
1
−
α
)

F
(
z
)

u(z) = {(1 - {\left | z \right |^2})^{n(1 - \alpha )}}F(z)

for some

F
(
z
)
∈
C
(

B
¯

)

F(z) \in C(\overline B )

. If for every

z
∈
B

z \in B

there corresponds an

r
(
z
)
:
0
>
r
(
z
)
>
1

r(z):0 > r(z) > 1

and an automorphism

ψ
z

{\psi _z}

with

ψ
z

(
0
)
=
z

{\psi _z}(0) = z

such that
\[

u
(
z
)
=

1

g
α

(
r
(
z
)
)

∫
S

u
∘

ψ
z

(
r
(
z
)
ζ
)
d
σ
(
ζ
)

,

u(z) = \frac {1}{{{g_\alpha }(r(z))}}\int _S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,

\]
then

Δ
~

u
(
z
)
=
−
4

n
2

α
(
1
−
α
)
u
(
z
)
,
z
∈
B

\tilde \Delta u(z) = - 4{n^2}\alpha (1 - \alpha )u(z),z \in B

. Here

Δ
~

\tilde \Delta

is the invariant Laplacian and

g
α

(
r
)

{g_\alpha }(r)

is the hypergeometric function

F
(
n
−
n
α
,
n
−
n
α
,
n
;

r
2

)

F(n - n\alpha ,n - n\alpha ,n;{r^2})

.
A generalization of the punctured neighborhood theorem
Authors: Lee Woo Young 
Published: June 2010

If

T
∈

L

(
X
)

T \in \mathcal {L}(X)

is regular on a Banach space

X
X

, with finite- dimensional intersection

T

−
1

(
0
)
∩
T
(
X
)

{T^{ - 1}}(0) \cap T(X)

, and if

S
,

S
′

S,\;S’

are invertible, commute with

T
T

and have sufficiently small norm, then

dim
⁡

(
T
−

S
′

)

−
1

(
0
)
=
dim
⁡

(
T
−
S

)

−
1

(
0
)

\dim {(T - S’)^{ - 1}}(0) = \dim {(T - S)^{ - 1}}(0)

and

dim
⁡
X

/

(
T
−

S
′

)
X
=
dim
⁡
X

/

(
T
−
S
)
X

\dim X/(T - S’)X = \dim X/(T - S)X

.