bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Bernstein-type inequality for the Jacobi polynomial
Authors: Chow Yunshyong , Gatteschi L. , Wong R. 
Published: June 2010

Let

P
n

(
α
,
β
)

(
x
)

P_n^{(\alpha ,\beta )}(x)

be the Jacobi polynomial of degree n. For

−

1
2

≤
α
,
β
≤

1
2

- \frac {1}{2} \leq \alpha ,\beta \leq \frac {1}{2}

, and

0
≤
θ
≤
π

0 \leq \theta \leq \pi

, it is proved that
\[

(
sin
⁡

θ
2

)

α
+

1
2

(
cos
⁡

θ
2

)

β
+

1
2

|

P
n

(
α
,
β
)

(
cos
⁡
θ
)

|

≤

Γ
(
q
+
1
)

Γ
(

1
2

)

(

n
+
q

n

)

N

−
q
−

1
2

,

{(\sin \frac {\theta }{2})^{\alpha + \frac {1}{2}}}{(\cos \frac {\theta }{2})^{\beta + \frac {1}{2}}}|P_n^{(\alpha ,\beta )}(\cos \theta )| \leq \frac {{\Gamma (q + 1)}}{{\Gamma (\frac {1}{2})}}\left ( {\begin {array}{*{20}{c}} {n + q} \\ n \\ \end {array} } \right ){N^{ - q - \frac {1}{2}}},

\]
where

q
=
max
(
α
,
β
)

q = \max (\alpha ,\beta )

and

N
=
n
+

1
2

(
α
+
β
+
1
)

N = n + \frac {1}{2}(\alpha + \beta + 1)

. When

α
=
β
=
0

\alpha = \beta = 0

, this reduces to a sharpened form of the well-known Bernstein inequality for the Legendre polynomial.
Some strict inclusions between spaces of 𝐿^{𝑝}-multipliers
Authors: Price J. F. 
Published: June 2010

Suppose that the Hausdorff topological group

G
G

is either compact or locally compact abelian and that

C
c

{C_c}

denotes the set of continuous complex-valued functions on

G
G

with compact supports. Let

L
p
q

L_p^q

denote the restrictions to

C
c

{C_c}

of the continuous linear operators from

L
p

(
G
)

{L^p}(G)

into

L
q

(
G
)

{L^q}(G)

which commute with all the right translation operators. When

1
≦
p
>
q
≦
2

1 \leqq p > q \leqq 2

or

2
≦
q
>
p
≦
∞

2 \leqq q > p \leqq \infty

it is known that
\[

(
1
)

L
p
p

⊂

L
q
q

.

(1)\quad L_p^p \subset L_q^q.

\]
The main result of this paper is that the inclusion in (1) is strict unless

G
G

is finite. In fact it will be shown, using a partly constructive proof, that when

G
G

is infinite
\[

⋃

1
≦
q
>
p

L
q
q

⫋

L
p
p

⫋

⋂

p
>
q
≦
2

L
q
q

\bigcup \limits _{1 \leqq q > p} {L_q^q \subsetneqq } L_p^p \subsetneqq \bigcap \limits _{p > q \leqq 2} {L_q^q}

\]
for

1
>
p
>
2

1 > p > 2

, with the first inclusion remaining strict when

p
=
2

p = 2

and the second inclusion remaining strict when

p
=
1

p = 1

. (Similar results also hold for

2
≦
p
≦
∞

2 \leqq p \leqq \infty

.) When

G
G

is compact, simple relations will also be developed between idempotent operators in

L
p
q

L_p^q

and lacunary subsets of the dual of

G
G

which will enable us to find necessary conditions so that inclusion (1) is strict even if, for example,

L
p
p

L_p^p

and

L
q
q

L_q^q

are replaced by the sets of idempotent operators in

L
p
p

L_p^p

and

L
q
q

L_q^q

respectively.
Analysis with weak trace ideals and the number of bound states of Schrödinger operators
Authors: Simon Barry 
Published: June 2010

We discuss interpolation theory for the operator ideals

I
p
w

I_p^w

defined on a separable Hilbert space as those operators A whose singular values

μ
n

(
A
)

{\mu _n}(A)

obey

μ
n

⩽
c

n

−
1

/

p

{\mu _n} \leqslant c{n^{ - 1/p}}

for some c. As an application we consider the functional

N
(
V
)
=
dim

N(V) = \dim

(spectral projection on

(
−
∞
,
0
)

( - \infty ,0)

for

−
Δ
+
V

- \Delta + V

) on functions V on

R

n

,
n
⩾
3

{{\mathbf {R}}^n},n \geqslant 3

. We prove that for any

ϵ
>
0
:
N
(
V
)
⩽

C
ϵ

(

‖
V
‖

n

/

2
+
ϵ

+

‖
V
‖

n

/

2
−
ϵ

)

n

/

2

\epsilon > 0:N(V) \leqslant C_\epsilon (\left \| V \right \|_{n/2 + \epsilon } + \left \| V \right \|_{n/2 - \epsilon })^{n/2}

where

‖
⋅
‖

p

{\left \| \cdot \right \|_p}

is an

L
p

{L^p}

norm and that

lim

λ
→
∞

N
(
λ
V
)

/

λ

n

/

2

=

(
2
π

)

−
n

τ
n

∫

|

V
−

(
x
)

|

n

/

2

d
n

x

{\lim \nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint |{V_ - }(x){|^{n/2}}{d^n}x

for any

V
∈

L

n

/

2
−

∩

L

n

/

2
+

V \in {L^{n/2 - }} \cap {L^{n/2 + }}

. Here

V
−

{V_ - }

is the negative part of V and

τ
n

{\tau _n}

is the volume of the unit ball in

R

n

{{\mathbf {R}}^n}

.
Some infinite free boundary problems
Authors: Tepper David E. , Wildenberg Gerald 
Published: June 2010

Let

Γ
\Gamma

be the boundary of an unbounded simply connected region

D

\mathcal {D}

, and let

C

(
Γ
)

\mathcal {C}(\Gamma )

denote the family of all simply connected regions

Δ
⊂

D

\Delta \subset \mathcal {D}

such that

∂
Δ
=
Γ
∪
γ

\partial \Delta = \Gamma \cup \gamma

where

γ
∩
Γ

\gamma \cap \Gamma

contains only the infinite point. For

Δ
∈

C

(
Γ
)

\Delta \in \mathcal {C}(\Gamma )

we call

γ
\gamma

the free boundary of

Δ
\Delta

. Given a positive constant

λ
\lambda

, we seek to find a region

Δ
λ

∈

C

(
Γ
)

{\Delta _\lambda } \in \mathcal {C}(\Gamma )

with free boundary

γ
λ

{\gamma _\lambda }

such that there is a bounded harmonic function V in

Δ
λ

{\Delta _\lambda }

with the properties that (i)

V
=
0

V = 0

on

Γ
\Gamma

, (ii)

V
=
1

V = 1

on

γ
\gamma

, (iii)

|

grad

V
(
z
)

|

=
λ

\left | {{\text {grad }}V(z)} \right | = \lambda

for

z
∈

γ
λ

z \in {\gamma _\lambda }

. We give sufficient conditions for existence and uniqueness of

Δ
λ

{\Delta _\lambda }

. We also give quantitative properties of

γ
λ

{\gamma _\lambda }

.
On meromorphic solutions of algebraic differential equations
Authors: Strelitz Sh. 
Published: June 2010

The Malmquist Theorem is generalized for equations of the type

R
(
z
,

w
,

w
′

,

…

,

w

(
n
)

)

=

P
(
z
,

w
)

/

Q
(
z
,

w
)

R(z,\,w,\,w’,\, \ldots \,,\,{w^{(n)}})\, = \,{{P(z,\,w)}/{Q(z,\,w)}}

where P, Q and R are polynomials of w and

w
,

w
′

,

…

,

w

(
n
)

w,\,w’,\, \ldots \,,\,{w^{(n)}}

respectively with meromorphic coefficients of finite order.
Weighted estimates for fractional powers of partial differential operators
Authors: Johnson Raymond 
Published: June 2010

It is shown that fractional powers defined by the wave polynomial

P
(
ξ
)
=

ξ

1

2

+
⋯
+

ξ
p
2

−

ξ

p
+
1

2

−
⋯
−

ξ
n
2

P(\xi ) = \xi _{^1}^2 + \cdots + \xi _p^2 - \xi _{p + 1}^2 - \cdots - \xi _n^2

, defined in terms of Fourier transforms by

T
λ

f

^

=

|

P
(
ξ
)

|

λ

f
^

\widehat {{T^\lambda }f} = {\left | {P(\xi )} \right |^\lambda }\hat f

, are in the Bernstein subalgebra of functions with integrable Fourier transforms for

λ
>
(
n
−
1
)

/

2

\lambda > (n - 1)/2

, provided

f
∈

C
c
m

f \in C_c^m

with

m
m

large enough. The proof uses embedding theorems for Besov spaces and Stein’s theorem on interpolation of analytic families of operators.
A remainder formula and limits of cardinal spline interpolants
Authors: Goodman T. N. T. , Lee S. L. 
Published: June 2010

A Peano-type remainder formula
\[

f
(
x
)
−

S
n

(
f
;

x
)
=

∫

−
∞

∞

K
n

(
x
,

t
)

f

(
n
+
1
)

(
t
)

d
t

f(x) - {S_n}(f;\,x) = \int _{ - \infty }^\infty {{K_n}(x,\,t){f^{(n + 1)}}(t)\,dt}

\]
for a class of symmetric cardinal interpolation problems C.I.P.

(
E
,

F
,

x

)

(E,\,F,\,{\mathbf {x}})

is obtained, from which we deduce the estimate

|

|

f
−

S

n
,
r

(
f
;

)

|

|

∞

⩽
K

|

|

f

(
n
+
1
)

|

|

∞

||f - {S_{n,r}}(f;\,)|{|_\infty } \leqslant K||{f^{(n + 1)}}|{|_\infty }

. It is found that the best constant

K
K

is obtained when

x

{\mathbf {x}}

comprises the zeros of the Euler-Chebyshev spline function. The remainder formula is also used to study the convergence of spline interpolants for a class of entire functions of exponential type and a class of almost periodic functions.
On the double suspension homomorphism at odd primes
Authors: Harper J. R. , Miller H. R. 
Published: June 2010

We work with the

E
1

{E_1}

-term for spheres and the stable Moore space, given by the

Λ
\Lambda

-algebra at odd primes. Writing

W
(
n
)
=
Λ
(
2
n
+
1
)

/

Λ
(
2
n
−
1
)

W(n) = \Lambda (2n + 1)/\Lambda (2n - 1)

and

M
(
0
)
=

H
∗

(

S
0

∪
p

e
1

)

M(0) = {H_ \ast }({S^0}{ \cup _p}{e^1})

, we construct compatible maps

f
n

⋅
W
(
n
)
→
M
(
0
)

⊗
~

Λ

{f_n} \cdot W(n) \to M(0)\tilde \otimes \Lambda

and prove the Metastability Theorem: in homology

f
n

{f_n}

induces an isomorphism for

σ
>
2
(

p
2

−
1
)
(
s
−
2
)
+
p
q
n
−
2
p
−
2

\sigma > 2({p^2} - 1)(s - 2) + pqn - 2p - 2

where

σ
=
stem degree

\sigma = \text {stem degree}

,

s
=
homological degree

s = \text {homological degree }

resulting from the bigrading of

Λ
\Lambda

and

q
=
2
p
−
2

q = 2p - 2

. There is an operator

υ
1

{\upsilon _1}

corresponding to the Adams stable self-map of the Moore space and

υ
1

{\upsilon _1}

extends to

W
(
n
)

W(n)

. A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map

f
n

{f_n}

induces an isomorphism on homology after inverting

υ
1

{\upsilon _1}

.
Weighted norm inequalities for the Fourier transform
Authors: Muckenhoupt Benjamin 
Published: June 2010

Given

p
p

and

q
q

satisfying

1
>
p
⩽
q
>
∞

1 > p \leqslant q > \infty

, sufficient conditions on nonnegative pairs of functions

U
,
V

U,V

are given to imply
\[

[

∫

R
n

|

f
^

(
x
)

|

q

U
(
x
)

d
x

]

1

/

q

⩽
c

[

∫

R
n

|

f
(
x
)

|

p

V
(
x
)

d
x

]

1

/

p

,

{\left [ {\int _{{R^n}}^{} {|\hat f(x){|^q}U(x)\,dx}} \right ]^{1/q}} \leqslant c{\left [ {\int _{{R^n}}^{} {|f(x){|^p}V(x)\,dx}} \right ]^{1/p}},

\]
where

f
^

\hat f

denotes the Fourier transform of

f
f

, and

c
c

is independent of

f
f

. For the case

q
=

p
′

q = p’

the sufficient condition is that for all positive

r
r

,
\[

[

∫

U
(
x
)
>
B
r

U
(
x
)

d
x

]

[

∫

V
(
x
)
>

r

p
−
1

V

(
x
)

−
1

/

(
p
−
1
)

d
x

]

⩽
A
,

\left [ {\int _{U(x) > Br} {U(x)\;dx}} \right ]\left [ {\int _{V(x) > {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right ] \leqslant A,

\]
where

A
A

and

B
B

are positive and independent of

r
r

. For

q
≠

p
′

q \ne p’

the condition is more complicated but also is invariant under rearrangements of

U
U

and

V
V

. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of

U
U

and

V
V

. Examples are given to show that the sufficient condition is not necessary for a pair

U
,
V

U,V

if the norm inequality is assumed only for that pair.
Almost everywhere summability on nilmanifolds
Authors: Hulanicki Andrzej , Jenkins Joe W. 
Published: June 2010

Let

G
G

be a stratified, nilpotent Lie group and let

L
L

be a homogeneous sublaplacian on

G
G

. Let

E
(
λ
)

E(\lambda )

denote the spectral resolution of

L
L

on

L
2

(
G
)

{L^2}(G)

. Given a function

K
K

on

R

+

\mathbf {R}^+

, define the operator

T
K

{T_K}

on

L
2

(
G
)

{L^2}(G)

by

T
k

f
=

∫
0
∞

K
(
λ
)

d
E
(
λ
)

f

{T_k}f = \int _0^\infty \, {K(\lambda )\;dE(\lambda )\,f}

. Sufficient conditions on

K
K

to imply that

T
K

{T_K}

is bounded on

L
1

(
G
)

{L^1}(G)

and the maximal operator

K

∗

φ
(
x
)
=

sup

t
>
0

|

T

K
t

φ
(
x
)

|

K^{\ast } \varphi (x) = \sup _{t > 0}|{T_{K_t}}\varphi (x)|

(where

K
t

(
λ
)
=
K
(
t
λ
)

{K_t}(\lambda ) = K(t\lambda )

) is of weak type

(
1
,
1
)

(1,1)

are given. Picking a basis

e
0

,

e
1

,
…

{e_0},{e_1},\ldots

of

L
2

(
G

/

Γ
)

{L^2}(G/\Gamma )

(

Γ
\Gamma

being a discrete cocompact subgroup of

G
G

) consisting of eigenfunctions of

L
L

, we obtain almost everywhere and norm convergence of various summability methods of

Σ
(
φ
,

e
j

)

e
j

,
φ
∈

L
p

(
G

/

Γ
)
,
1
⩽
p
>
∞

\Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p > \infty

.
Diffusion dependence of the FitzHugh-Nagumo equations
Authors: Collins Clyde 
Published: June 2010

We investigate the behavior of the solutions of
\[

u
t

=

u

x

x

−
α

u
−
v
+
f
(
u
)
,

v
t

=
η

v

x

x

+
σ

u
−
γ
v
,

\begin {array}{*{20}{c}} {{u_t} = {u_{x\,x}} - \alpha \,u - v + f(u),} \\ {{v_t} = \eta \,{v_{x\,x}} + \sigma \,u - \gamma v,} \\ \end {array}

\]
as

η
\eta

tends to zero from above.
On maximal rearrangement inequalities for the Fourier transform
Authors: Jurkat W. B. , Sampson G. 
Published: June 2010

Suppose that

w
w

is a measurable function on

R

n

{{\mathbf {R}}^n}

and denote by

W
=

w
∗

W = {w^ \ast }

the decreasing rearrangement of

|
w
|

\left | w \right |

(provided that it exists). We show that the

n
n

-dimensional Fourier transform

f
^

\hat f

satisfies (1)
\[

‖

w

f
^

‖

q

⩽

‖

W

(

f
^

)

∗

‖

q

⩽
C

‖

W
(
t
)

∫
0

1

/

t

f
∗

‖

(
C

absolute constant

)
,

{\left \| {w\hat f} \right \|_q} \leqslant {\left \| {W{{(\hat f)}^ \ast }} \right \|_q} \leqslant C\left \| {W(t)\int _0^{1/t} {{f^ \ast }} } \right \|\quad (C\ {\text {absolute constant}}),

\]
if

1
>
q
>
∞

1 > q > \infty

and

t

2

/

q
−
1

W
(
t
)
↘

{t^{2/q - 1}}W(t) \searrow

for

t
>
0

t > 0

. We also show that (2)
\[

‖

w

f
^

‖

q

⩾

c

n
,
q

‖

w
(
x
)

∫

|
v
|

⩽
1

/

|
x
|

f
(
y
)

d
y

‖

q

(
f

nonnegative),

{\left \| {w\hat f} \right \|_q} \geqslant {c_{n,q}}{\left \| {w(x)\int _{\left | v \right | \leqslant 1/\left | x \right |} {f(y)} dy} \right \|_q}\quad (f\ {\text {nonnegative),}}

\]
if

1
>
q
>
∞

1 > q > \infty

and

w
w

is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if

f
f

is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of

W
W

and

f
∗

{f^ \ast }

. The cases

q
≠
2

q \ne 2

of (1) can be derived from the case

q
=
2

q = 2

(and same

f
f

) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if

q
⩾
2

q \geqslant 2

(then the extra condition on

W
W

is automatically satisfied).
The role of countable dimensionality in the theory of cell-like relations
Authors: Ancel Fredric D. 
Published: June 2010

Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact

U

V
∞

U{V^\infty }

point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map

f
:
X
→
Y

f:X \to Y

is a hereditary shape equivalence if there is a sequence

{

K
n

}

\{ {K_n}\}

of closed subsets of

Y
Y

such that (1)

Y
−

⋃

n
=
1

∞

K
n

Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}}

is countable dimensional, and (2)

f

|

f

−
1

(

K
n

)
:

f

−
1

(

K
n

)
→

K
n

f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}

is a hereditary shape equivalence for each

n
≥
1

n \geq 1

. Theorem 5.9. If

f
:
X
→
Y

f:X \to Y

is a proper onto map whose point inverses are

U

V
∞

U{V^\infty }

sets, then

Y
Y

is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if

Y
Y

is countable dimensional, then

Y
Y

is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.
General defect relations of holomorphic curves
Authors: Niino Kiyoshi 
Published: June 2010

Let

x
:

C

→

P
n

C

x:{\mathbf {C}} \to {P_n}{\mathbf {C}}

be a holomorphic curve of finite lower order

μ
\mu

, and let

A
=
{
α
}

A = \{ \alpha \}

be an arbitrary finite family of holomorphic curves

α
:

C

→

(

P
n

C

)
∗

\alpha :{\mathbf {C}} \to {({P_n}{\mathbf {C}})^\ast }

satisfying

T
(
r
,
α
)
=
o
(
T
(
r
,
x
)
)

(
r
→
∞
)

T(r,\alpha ) = o(T(r,x))\;(r \to \infty )

. Suppose

x
x

is nondegenerate with respect to

A
A

, and

A
A

is in general position. We show the following general defect relations: (1)

x
x

has at most

n
n

deficient curves in

A
A

if

μ
=
0

\mu = 0

. (2)

∑

α
∈
A

δ
(
α
)
≤
n

if

0
>
μ
≤
1

/

2

\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq n\;{\text {if}}\;0 > \mu \leq 1/2}

. (3)

∑

α
∈
A

δ
(
α
)
≤
[
2
n
μ
]
+
1

if

1

/

2
>
μ
>
+
∞

\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text {if}}\;1/2 > \mu > + \infty }

.
The law of the iterated logarithm in uniformly convex Banach spaces
Authors: Ledoux Michel 
Published: June 2010

We give necessary and sufficient conditions for a random variable

X
X

with values in a uniformly convex Banach space

B
B

to satisfy the law of the iterated logarithm. Precisely, we show that a

B
B

-valued random variable

X
X

satisfies the (compact) law of the iterated logarithm if and only if

E
{

|

|

X

|

|

2

/

L
2

|

|

X

|

|

}
>
∞

E\{ ||X|{|^2}/{L_2}||X||\} > \infty

, the family

{

|

x

∗

(
X
)

|

2

;

x

∗

∈

B

∗

,

|

|

x

∗

|

|

=
1
}

\{ |{x^{\ast }}(X){|^2};\,{x^{\ast }} \in {B^{\ast }},\,||{x^{\ast }}|| = 1\}

is uniformly integrable and

S
n

/

a
n

→
0

{S_n}/{a_n} \to 0

in probability.
Bifurcation phenomena associated to the 𝑝-Laplace operator
Authors: Guedda Mohammed , Véron Laurent 
Published: June 2010

We determine the structure of the set of the solutions

u
u

of

−

(

|

u
x

|

p
−
2

u
x

)
x

+
f
(
u
)
=
λ

|

u

|

p
−
2

u

- {(|{u_x}{|^{p - 2}}{u_x})_x} + f(u) = \lambda |u{|^{p - 2}}u

on

(
0
,

1
)

(0,\,1)

such that

u
(
0
)
=
u
(
1
)
=
0

u(0) = u(1) = 0

, where

p
>
1

p > 1

and

λ
∈

R

\lambda \in {\mathbf {R}}

. We prove that the solutions with

k
k

zeros are unique when

1
>
p
⩽
2

1 > p \leqslant 2

but may not be so when

p
>
2

p > 2

.
Note on osculatory rational interpolation
Authors: Salzer Herbert E. 
Published: June 2010

In n-point osculatory interpolation of order

r
i

−

1

{r_i}\, - \,1

at points

x
i

{x_i}

,

i

=

1
,

2
,

⋯

,

n
,

i\, = \,1,\,2,\, \cdots \,,\,n,

, by a rational expression

N
(
x
)

/

D
(
x
)

{N(x)}/{D(x)}

, where

N
(
x
)

N(x)

and

D
(
x
)

D(x)

are polynomials

∑

a
j

x
j

\sum {{a_j}{x^j}}

and

∑

b
j

x
j

\sum {{b_j}{x^j}}

, We use the lemma that the system (1)

{
N
(

x

i

)

/

D
(

x

i

)
}

(
m
)

=

f

(
m
)

(

x

i

)
,

m
=
0
,
1
,
⋯
,

r

i

−
1

{{\{N({{x}_{i}})/D({{x}_{i}})\}}^{(m)}}\,=\,{{f}^{(m)}}({{x}_{i}}),\,m=0,1,\cdots ,{{r}_{i}}-1

, is equivalent to (2)

N

(
m
)

(

x
i

)

=

{

f
(

x
i

)
D
(

x
i

)

}

(
m
)

,

m

=

0
,

1
,

⋯

,

r
i

−

1
,

D
(

x
i

)

≠

0

{N^{(m)}}({{x_i}})\, = \,{\{{f({{x_i}})D({{x_i}})}\}^{(m)}},\,m\, = \,0,\,1,\, \cdots \,,\,{r_i}\, - \,1,\,D({{x_i}})\, \ne \,0

. This equivalence does not require

N
(
x
)

N(x)

or

D
(
x
)

D(x)

to be a polynomial or even a linear combination of given functions. The lemma implies that (1), superficially non-linear in

a
j

{a_j}

and

b
j

{b_j}

, being the same as (2), is actually linear. For the n-point interpolation problem, the linear system, of order

∑

i
=
1

n

r
i

\sum \limits _{i = 1}^n {{r_i}}

, which might be large, is replaceable by separate linear Systems of orders

r
i

{r_i}

(or even

r
i

+

r

i
+
1

+

⋯

+

r

i
+
j

{r_i}\, + \,{r_{i + 1}}\, + \, \cdots \, + \,{r_{i + j}}

when conveniently small) by applying the lemma to the continued fraction (3)

N
(
x
)

/

D
(
x
)

=

a

1
,
0

+

x
−

x

1

|

|

a

1
,
1

+

x
−

x

1

|

|

a

1
,
2

+

⋯

+

x
−

x

1

|

|

a

1
,

r

1

−
1

+

x
−

x

1

|

|

a

2
,
0

+

x
−

x

1

|

|

a

2
,
1

+

⋯

+

x
−

x

1

|

|

a

2
,

r

2

−
1

+

x
−

x

1

|

|

a

3
,
0

+

⋯

+

x
−

x

n
−
1

|

|

a

n
,
0

+

x
−

x

n

|

|

a

n
,
1

+

⋯

+

x
−

x

n

|

|

a

n
,

r

n

−
1

{N(x)}/{D(x)\,=\,{{a}_{1,0}}\,+\,\tfrac {x- {{x}_{1}}|}{| {{a}_{1,1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{1,2}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{1,{{r}_{1}}-1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,0}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,1}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,{{r}_{2}}-1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{3,0}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{n-1}} |}{| {{a}_{n,0}} }\,+\,\tfrac {x-\ {{x}_{n}} |}{| {{a}_{n,1}} }\,+\,\cdots \,+\tfrac {x- {{x}_{n}} |}{| {{a}_{n,{{r}_{n}}-1}} }}\;

. In (3), which has the property (proven in two ways) that the determination of

a

i
,
m

{a_{i,m}}

is independent of all a’s that follow, we find

a

i
,
m

{a_{i,m}}

stepwise, but several at a time (instead of singly which is more tedious), retrieving them readily from the solutions of those lower-order linear systems.
A method for the computation of the error function of a complex variable
Authors: Strand Otto Neall 
Published: June 2010

This paper presents a method of computing

z
≡

(

2

/

π

)

∫
0
z

e

−

u
2

d
u

z \equiv \left ( {2/\sqrt \pi } \right )\int _0^z {{e^{ - {u^2}}}du}

, where

z
z

is complex. It is shown that

z
≡
1
−

erf

z

z \equiv 1 - {\text {erf }}z

has no zeros in the right-hand half plane. An estimate of

|

erfc

z

|

|{\text {erfc }}z|

is derived.
A generalization of a class of test matrices
Authors: Herbold Robert J. 
Published: June 2010

We consider matrices of the following form:

G
n

(

a
1

,

a
2

,
⋯
,

a

n
−
1

,

b
1

,

b
2

⋯

b
n

)
=

{G_n}({a_1},{a_2}, \cdots ,{a_{n - 1}},{b_1},{b_2} \cdots {b_n}) =

(

β

i
,
j

)
,
1
≦
i

({\beta _{i,j}}),1 \leqq i

,

j
≦
n

j \leqq n

, where

a
1

,
⋯
,

a

n
−
1

,

b
1

,
⋯
,

b
n

{a_1}, \cdots ,{a_{n - 1}},{b_1}, \cdots ,{b_n}

are constants and
\[

β
i

,
j

=

b
j

,

j
≧
i
;

β

i
j

=

a
j

,

j
>
i
.

{\beta _i}_{,j} = {b_j},{\text { }}j \geqq i;{\text { }}{\beta _{ij}} = {a_j},{\text { }}j > i.

\]
We deduce in analytic form the determinant, inverse matrix, characteristic equation, and eigenvectors of

G
n

{G_n}

. Knowing these properties enables us to generate valuable test matrices by appropriately selecting the order and elements of

G
n

{G_n}

.
On the construction of Gaussian quadrature rules from modified moments.
Authors: Gautschi Walter 
Published: June 2010

Given a weight function

ω
(
x
)

\omega (x)

on

(
α
,
β
)

(\alpha ,\beta )

, and a system of polynomials

{

p
k

(
x
)

}

k
=
0

∞

\left \{ {{p_k}(x)} \right \}_{k = 0}^\infty

, with degree

p
k

(
x
)
=
k

{p_k}(x) = k

, we consider the problem of constructing Gaussian quadrature rules

∫
α
β

f
(
x
)
ω
(
x
)
d
x
=

∑

r
=
1

n

λ
r

(
n
)

f
(

ξ
r

(
n
)

)

\int _\alpha ^\beta {f(x)\omega (x)dx = \sum \nolimits _{r = 1}^n {{\lambda _r}^{(n)}f({\xi _r}^{(n)})} }

from "modified moments"

v
k

=

∫
α
β

p
k

(
x
)
ω
(
x
)
d
x

{v_k} = \int _\alpha ^\beta {{p_k}(x)\omega (x)dx}

. Classical procedures take

p
k

(
x
)
=

x
k

{p_k}(x) = {x^k}

, but suffer from progressive ill-conditioning as

n
n

increases. A more recent procedure, due to Sack and Donovan, takes for

{

p
k

(
x
)
}

\{ {p_k}(x)\}

a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals

[
α
,
β
]

[\alpha ,\beta ]

. In support of this observation, we obtain upper bounds for the respective asymptotic condition number. In special cases, these bounds grow like a fixed power of

n
n

. We also derive an algorithm for solving the problem considered, which generalizes one due to Golub and Welsch. Finally, some numerical examples are presented.