aarjaneiro/planetmath huggingface dataset:
wavelet set

wavelet set

Definition

An

(

orthonormal

dyadic)

wavelet set

on

ℝ

is a subset

E

⊂

ℝ

such that

1.

χ

E

∈

L

2

⁢

(

ℝ

)

(since

∥

χ

E

∥

=

m

⁢

(

E

)

, this implies

m

⁢

(

E

)

<

∞

).

2.

χ

E

m

⁢

(

E

)

is the Fourier transform of an

orthonormal dyadic wavelet

,

where

χ

E

is the

characteristic function

of

E

, and

m

⁢

(

E

)

is the

Lebesgue measure

of

E

.

Characterization

E

⊂

ℝ

is a wavelet set iff

1.

{

E

+

2

⁢

π

⁢

n

}

n

∈

ℤ

is a

measurable partition

of

ℝ

; i.e.

ℝ

\

⋃

n

∈

ℤ

{

E

+

2

⁢

π

⁢

n

}

has

measure zero

, and

⋂

n

=

i

,

j

{

E

+

2

⁢

π

⁢

n

}

has measure zero if

i

≠

j

. In short,

E

is a

2

⁢

π

-translation “tiler” of

ℝ

2.

{

2

n

⁢

E

}

n

∈

ℤ

is a

2

-dilation “tiler” of

ℝ

(once again modulo sets of measure zero).

Notes

There are higher dimensional analogues to wavelet sets in

ℝ

, corresponding to wavelets in higher

dimensions

. Wavelet sets can be used to derive wavelets— by creating a set

E

satisfying the conditions given above, and using the inverse Fourier transform on

χ

E

, you are guaranteed to recover a wavelet. A particularly interesting open question is: do all wavelets contain wavelet sets in their frequency

support

?

Title

wavelet set

Canonical name

WaveletSet

Date of creation

2013-03-22 14:27:10

Last modified on

2013-03-22 14:27:10

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

7

Author

swiftset (1337)

Entry type

Definition

Classification

msc 46C99

Classification

msc 65T60

Related topic

wavelet

Related topic

Wavelet

Generated on Fri Feb 9 21:39:37 2018 by

LaTeXML
Ascoli-Arzelà theorem

Ascoli-Arzelà theorem

Let

Ω

be a

bounded

subset of

ℝ

n

and

(

f

k

)

a

sequence

of functions

f

k

:

Ω

→

ℝ

m

. If

{

f

k

}

is

equibounded

and

uniformly equicontinuous

then there exists a

uniformly convergent

subsequence

(

f

k

j

)

.

A more abstract (and more general) version is the following.

Let

X

and

Y

be

totally bounded

metric spaces

and let

F

⊂

𝒞

⁢

(

X

,

Y

)

be an uniformly equicontinuous family of

continuous mappings

from

X

to

Y

.
Then

F

is totally bounded (with respect to the

uniform convergence

metric induced by

𝒞

⁢

(

X

,

Y

)

).

Notice that the first version is a consequence of the second.
Recall, in fact, that a subset of a

complete metric space

is totally bounded if and only if its

closure

is

compact

(or

sequentially compact

).
Hence

Ω

is totally bounded and all the functions

f

k

have image in a totally bounded set. Being

F

=

{

f

k

}

totally bounded means that

F

¯

is sequentially compact and hence

(

f

k

)

has a

convergent

subsequence.

Title

Ascoli-Arzelà theorem

Canonical name

AscoliArzelaTheorem

Date of creation

2013-03-22 12:41:00

Last modified on

2013-03-22 12:41:00

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

13

Author

paolini (1187)

Entry type

Theorem

Classification

msc 46E15

Synonym

Arzelà-Ascoli theorem

Related topic

MontelsTheorem

Generated on Fri Feb 9 21:39:39 2018 by

LaTeXML
proof of Ascoli-Arzelà theorem

proof of Ascoli-Arzelà theorem

Given

ϵ

>

0

we aim at finding a

4

⁢

ϵ

-net in

F

i.e. a

finite set

of points

F

ϵ

such that

⋃

f

∈

F

ϵ

B

4

⁢

ϵ

⁢

(

f

)

⊃

F

(see the definition of

totally bounded

).
Let

δ

>

0

be given with respect to

ϵ

in the definition
of equi-continuity (see

uniformly equicontinuous

) of

F

.
Let

X

δ

be a

δ

-

lattice

in

X

and

Y

ϵ

be a

ϵ

-lattice in

Y

.
Let now

Y

ϵ

X

δ

be the set of

functions

from

X

δ

to

Y

ϵ

and define

G

ϵ

⊂

Y

ϵ

X

δ

by

G

ϵ

=

{

g

∈

Y

ϵ

X

δ

:

∃

f

∈

F

⁢

∀

x

∈

X

δ

d

⁢

(

f

⁢

(

x

)

,

g

⁢

(

x

)

)

<

ϵ

}

.

Since

Y

ϵ

X

δ

is a finite set,

G

ϵ

is finite too: say

G

ϵ

=

{

g

1

,

…

,

g

N

}

.
Then define

F

ϵ

⊂

F

,

F

ϵ

=

{

f

1

,

…

,

f

N

}

where

f

k

:

X

→

Y

is a function in

F

such that

d

⁢

(

f

k

⁢

(

x

)

,

g

k

⁢

(

x

)

)

<

ϵ

for all

x

∈

X

δ

(the

existence

of
such a function is guaranteed by the definition of

G

ϵ

).

We now will prove that

F

ϵ

is a

4

⁢

ϵ

-lattice in

F

.
Given

f

∈

F

choose

g

∈

Y

ϵ

X

δ

such that for all

x

∈

X

δ

it holds

d

⁢

(

f

⁢

(

x

)

,

g

⁢

(

x

)

)

<

ϵ

(this is possible as for all

x

∈

X

δ

there exists

y

∈

Y

ϵ

with

d

⁢

(

f

⁢

(

x

)

,

y

)

<

ϵ

).
We conclude that

g

∈

G

ϵ

and hence

g

=

g

k

for some

k

∈

{

1

,

…

,

N

}

. Notice also that for all

x

∈

X

δ

we
have

d

⁢

(

f

⁢

(

x

)

,

f

k

⁢

(

x

)

)

≤

d

⁢

(

f

⁢

(

x

)

,

g

k

⁢

(

x

)

)

+

d

⁢

(

g

k

⁢

(

x

)

,

f

k

⁢

(

x

)

)

<

2

⁢

ϵ

.

Given any

x

∈

X

we know that there exists

x

δ

∈

X

δ

such that

d

⁢

(

x

,

x

δ

)

<

δ

.
So, by

equicontinuity

of

F

,

d

⁢

(

f

⁢

(

x

)

,

f

k

⁢

(

x

)

)

≤

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

x

δ

)

)

+

d

⁢

(

f

k

⁢

(

x

)

,

f

k

⁢

(

x

δ

)

)

+

d

⁢

(

f

⁢

(

x

δ

)

,

f

k

⁢

(

x

δ

)

)

<

4

⁢

ϵ

.

Title

proof of Ascoli-Arzelà

theorem

Canonical name

ProofOfAscoliArzelaTheorem

Date of creation

2013-03-22 13:16:19

Last modified on

2013-03-22 13:16:19

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

12

Author

paolini (1187)

Entry type

Proof

Classification

msc 46E15

Generated on Fri Feb 9 21:39:41 2018 by

LaTeXML
Stone-Weierstrass theorem

Stone-Weierstrass theorem

Let

X

be a

compact space

and let

C

0

⁢

(

X

,

ℝ

)

be the algebra of

continuous

real functions
defined over

X

. Let

𝒜

be a

subalgebra

of

C

0

⁢

(

X

,

ℝ

)

for which the following
conditions hold:

1.

∀

x

,

y

∈

X

,

x

≠

y

,

∃

f

∈

𝒜

:

f

(

x

)

≠

f

(

y

)

2.

1

∈

𝒜

Then

𝒜

is dense in

C

0

⁢

(

X

,

ℝ

)

.

This

theorem

is a

generalization

of the classical

Weierstrass approximation theorem

to general spaces.

Title

Stone-Weierstrass theorem

Canonical name

StoneWeierstrassTheorem

Date of creation

2013-03-22 12:42:06

Last modified on

2013-03-22 12:42:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 46E15

Generated on Fri Feb 9 21:39:47 2018 by

LaTeXML
Laplace transform of Dirac delta

Laplace transform of Dirac delta

The Dirac delta (

http://planetmath.org/DiracDeltaFunction

)

δ

can be interpreted as a

linear functional

, i.e. a

linear mapping

from a

function space

, consisting e.g. of certain

real functions

, to

ℝ

(or

ℂ

), having the property

δ

⁢

[

f

]

=

f

⁢

(

0

)

.

One may think this as the

inner product

⟨

f

,

δ

⟩

=

∫

0

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

)

⁢

𝑑

t

of a

function

f

and another “function”

δ

, when the well-known

∫

0

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

0

)

is true.  Applying this to

f

⁢

(

t

)

:=

e

-

s

⁢

t

,  one gets

∫

0

∞

e

-

s

⁢

t

⁢

δ

⁢

(

t

)

⁢

𝑑

t

=

e

-

0

,

i.e. the

Laplace transform

ℒ

⁢

{

δ

⁢

(

t

)

}

=

1

.

(1)

By the

delay theorem

, this result may be generalised to

ℒ

{

δ

(

t

-

a

)

)

}

=

e

-

a

⁢

s

.

When introducing some “nascent

Dirac delta function

”, for example

η

ε

⁢

(

t

)

:=

{

1

ε

for

⁢

0

≤

t

≤

ε

,

0

for

t

>

ε

,

as an “approximation” of Dirac delta, we obtain the Laplace transform

ℒ

⁢

{

η

ε

⁢

(

t

)

}

=

∫

0

∞

e

-

s

⁢

t

⁢

η

ε

⁢

(

t

)

⁢

𝑑

t

=

∫

0

ε

e

-

s

⁢

t

ε

⁢

𝑑

t

+

∫

ε

∞

e

-

s

⁢

t

⋅

0

⁢

𝑑

t

=

1

ε

⁢

∫

0

ε

e

-

s

⁢

t

⁢

𝑑

t

=

1

-

e

-

ε

⁢

s

ε

⁢

s

.

As the

Taylor expansion

shows, we then have

lim

ε

→

0

+

⁡

ℒ

⁢

{

η

ε

⁢

(

t

)

}

=

1

,

being in accordance with (1).

Title

Laplace transform of Dirac delta

Canonical name

LaplaceTransformOfDiracDelta

Date of creation

2013-03-22 19:10:56

Last modified on

2013-03-22 19:10:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Result

Classification

msc 46E20

Classification

msc 44A10

Classification

msc 34L40

Generated on Fri Feb 9 21:39:49 2018 by

LaTeXML
conjugate index

conjugate index

For

p

,

q

∈

ℝ

,

1

<

p

,

q

<

∞

we say

p

and

q

are

if

1

p

+

1

q

=

1

. Formally, we will also define

q

=

∞

as conjugate to

p

=

1

and vice versa.

Conjugate indices

are used in the Hölder

inequality

(

http://planetmath.org/HolderInequality

) and more generally to define

conjugate spaces

.

Title

conjugate index

Canonical

name

ConjugateIndex

Date of creation

2013-03-22 12:21:35

Last modified on

2013-03-22 12:21:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46E30

Synonym

conjugate indices

Synonym

conjugate indexes

Synonym

conjugate exponent

Synonym

Holder exponent

Synonym

Hölder

exponent

Related topic

HolderInequality

Related topic

BeattysTheorem

Related topic

BoundedLinearFunctionalsOnLpmu

Generated on Fri Feb 9 21:39:54 2018 by

LaTeXML
generalization of Young inequality

generalization of Young inequality

It’s straightforward to extend

Young inequality

(

http://planetmath.org/YoungInequality

)
to an arbitrary finite number of : provided that

a

i

>

0

,

c

i

>

0

and

∑

i

=

1

n

1

c

i

=

1

r

,

(

∏

i

=

1

n

a

i

)

r

≤

r

⁢

∑

i

=

1

n

a

i

c

i

c

i

In fact,

(

∏

i

=

1

n

a

i

)

r

=

exp

[

log

(

∏

i

=

1

n

a

i

)

r

]

=

exp

⁡

[

r

⁢

∑

i

=

1

n

log

⁡

a

i

]

=

exp

⁡

[

r

⁢

∑

i

=

1

n

1

c

i

⁢

log

⁡

(

a

i

c

i

)

]

=

exp

⁡

[

∑

i

=

1

n

1

c

i

⁢

log

⁡

(

a

i

c

i

)

1

r

]

(by Jensen’s

inequality

and monotonicity of exp)

≤

exp

⁡

[

log

⁡

(

∑

i

=

1

n

1

c

i

⁢

a

i

c

i

1

r

)

]

=

r

⁢

∑

i

=

1

n

a

i

c

i

c

i

Remark: in the case

1

c

i

=

1

⁢

⁢

∀

i

one obtains:

(

∏

i

=

1

n

a

i

)

1

n

≤

1

n

⁢

∑

i

=

1

n

a

i

that is, the usual

arithmetic-geometric mean inequality

, which suggests
Young inequality could be regarded as a generalization of this classical result.
Actually, let’s consider the following restatement of
Young inequality. Having defined:

w

i

=

1

c

i

,

∑

i

=

1

n

w

i

=

W

=

1

r

,

x

i

=

a

i

1

w

i

we have:

(

∏

i

=

1

n

x

i

w

i

)

1

W

≤

1

W

⁢

∑

i

=

1

n

w

i

⁢

x

i

This

expression

shows that Young inequality is nothing else than
geometric-arithmentic

weighted

mean (

http://planetmath.org/ArithmeticMean

) inequality.

Title

generalization of Young inequality

Canonical

name

GeneralizationOfYoungInequality

Date of creation

2013-03-22 15:43:08

Last modified on

2013-03-22 15:43:08

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

25

Author

Andrea Ambrosio (7332)

Entry type

Result

Classification

msc 46E30

Generated on Fri Feb 9 21:39:56 2018 by

LaTeXML
generalized Hölder inequality

generalized Hölder inequality

Theorem

Let

1

≤

r

<

∞

and

1

≤

p

j

<

∞

, where

∑

j

=

1

n

1

p

j

=

1

r

. If

f

j

∈

L

p

j

for

1

≤

j

≤

n

, then

∏

j

=

1

n

f

j

∈

L

r

and

||

∏

j

=

1

n

f

j

||

r

≤

∏

j

=

1

n

||

f

j

||

p

j

.

The usual Hölder

inequality

has

n

=

2

and

r

=

1

.

Let

X

be a finite set, say

X

=

{

x

1

,

…

,

x

m

}

and

μ

is the

counting measure

on

X

, so that

μ

⁢

(

{

x

i

}

)

=

1

for all

i

. Let

f

j

⁢

(

x

i

)

=

a

i

⁢

j

≥

0

for

j

=

1

,

…

,

n

and take

r

=

1

. Then the inequality becomes:

∑

i

=

1

m

∏

j

=

1

n

a

i

⁢

j

≤

∏

j

=

1

n

(

∑

i

=

1

m

a

i

⁢

j

p

j

)

1

p

j

.

.

Now let

α

j

=

1

p

j

, and

b

i

⁢

j

=

a

i

⁢

j

p

j

. Then the
inequality becomes:

∑

i

=

1

m

∏

j

=

1

n

b

i

⁢

j

α

j

≤

∏

j

=

1

n

(

∑

i

=

1

m

b

i

⁢

j

)

α

j

.

Title

generalized Hölder inequality

Canonical name

GeneralizedHolderInequality

Date of creation

2013-03-22 16:54:35

Last modified on

2013-03-22 16:54:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46E30

Generated on Fri Feb 9 21:39:59 2018 by

LaTeXML
proof of bounded linear functionals on L^p⁢(μ)

proof of bounded linear functionals on

L

p

⁢

(

μ

)

If

(

X

,

𝔐

,

μ

)

is a

σ

-finite measure-space and

p

,

q

are Hölder

conjugates

(

http://planetmath.org/ConjugateIndex

) with

p

<

∞

, then we show that

L

q

is

isometrically isomorphic

to the

dual space

of

L

p

.

For any

g

∈

L

q

, define the

linear map

Φ

g

:

L

p

→

ℂ

,

f

↦

Φ

g

⁢

(

f

)

=

∫

f

⁢

g

⁢

𝑑

μ

.

This is a

bounded linear map

with

operator norm

∥

Φ

g

∥

=

∥

g

∥

q

(see

L

p

-norm is dual to

L

q

(

http://planetmath.org/LpNormIsDualToLq

)), so the map

g

↦

Φ

g

gives an

isometric embedding

from

L

q

to the dual space of

L

p

. It only remains to show that it is onto.

So, suppose that

Φ

:

L

p

→

ℂ

is a bounded linear map. It needs to be shown that there is a

g

∈

L

q

with

Φ

=

Φ

g

.
As any

σ

-finite measure is

equivalent

to a probability measure (

http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure

), there is a

bounded

h

>

0

such that

∫

h

⁢

𝑑

μ

=

1

.
Let

Φ

~

:

L

∞

→

ℂ

be the bounded linear map given by

Φ

~

⁢

(

f

)

=

Φ

⁢

(

h

⁢

f

)

. Then, there is a

g

0

∈

L

1

such that

Φ

⁢

(

h

⁢

f

)

=

Φ

~

⁢

(

f

)

=

∫

f

⁢

g

0

⁢

𝑑

μ

for every

f

∈

L

∞

(see bounded

linear functionals

on

L

∞

(

http://planetmath.org/BoundedLinearFunctionalsOnLinftymu

)). Set

g

=

h

-

1

⁢

g

0

and, for any

f

∈

L

p

, let

f

n

be the

sequence

f

n

=

f

⁢

1

{

|

h

-

1

f

|

<

n

}

.

As

h

-

1

⁢

f

n

∈

L

∞

,

∥

f

n

⁢

g

∥

1

=

∥

h

-

1

⁢

f

n

⁢

g

0

∥

1

=

Φ

⁢

(

sign

⁡

(

f

⁢

g

0

)

⁢

f

n

)

≤

∥

Φ

∥

⁢

∥

f

n

∥

p

.

Letting

n

tend to

infinity

, dominated convergence says that

f

n

→

f

in the

L

p

-norm, so Fatou’s lemma gives

∥

f

⁢

g

∥

1

≤

lim inf

n

→

∞

⁡

∥

f

n

⁢

g

∥

1

≤

∥

Φ

∥

⁢

∥

f

∥

p

.

In particular,

∥

g

∥

q

≤

∥

Φ

∥

(see

L

p

-norm is dual to

L

q

(

http://planetmath.org/LpNormIsDualToLq

)), so

g

∈

L

q

. As

|

f

n

⁢

g

|

≤

|

f

⁢

g

|

are in

L

1

, dominated convergence finally gives

∫

f

⁢

g

⁢

𝑑

μ

=

lim

n

→

∞

⁡

∫

f

n

⁢

g

⁢

𝑑

μ

=

lim

n

→

∞

⁡

Φ

⁢

(

f

n

)

=

Φ

⁢

(

f

)

so

Φ

g

=

Φ

as required.

Title

proof of bounded linear functionals on

L

p

⁢

(

μ

)

Canonical name

ProofOfBoundedLinearFunctionalsOnLpmu

Date of creation

2013-03-22 18:38:19

Last modified on

2013-03-22 18:38:19

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 46E30

Classification

msc 28A25

Generated on Fri Feb 9 21:40:02 2018 by

LaTeXML
proof of L^p-norm is dual to L^q

proof of

L

p

-norm is dual to

L

q

Let

(

X

,

𝔐

,

μ

)

be a

σ

-

finite measure space

and

p

,

q

be Hölder conjugates. Then, we show that a

measurable function

f

:

X

→

ℝ

has

L

p

-norm

∥

f

∥

p

=

sup

⁡

{

∥

f

⁢

g

∥

1

:

g

∈

L

q

,

∥

g

∥

q

=

1

}

.

(1)

Furthermore, if either

p

<

∞

and

∥

f

∥

p

<

∞

or

p

=

1

then

μ

is not required to be

σ

-finite.

If

∥

f

∥

p

=

0

then

f

is zero almost everywhere, and both sides of equality (

1

) are zero. So, we only need to consider the case where

∥

f

∥

p

>

0

.

Let

K

be the right hand side of equality (

1

).
For any

g

∈

L

q

with

∥

g

∥

q

=

1

, the Hölder

inequality

gives

∥

f

⁢

g

∥

1

≤

∥

f

∥

p

, so

K

≤

∥

f

∥

p

. Only the reverse inequality remains to be shown.

If

1

<

p

<

∞

and

∥

f

∥

p

<

∞

then, setting

g

=

|

f

|

p

-

1

gives

∥

g

∥

q

=

(

∫

|

f

|

p

⁢

𝑑

μ

)

1

q

=

∥

f

∥

p

p

-

1

<

∞

.

Therefore,

g

∈

L

q

and,

K

≥

∥

f

⁢

(

g

/

∥

g

∥

q

)

∥

1

=

∥

|

f

|

p

∥

1

/

∥

g

∥

q

=

∥

f

∥

p

p

/

∥

f

∥

p

p

-

1

=

∥

f

∥

p

.

On the other hand, if

p

=

1

so that

q

=

∞

, then setting

g

=

1

gives

∥

g

∥

q

=

1

and

K

≥

∥

f

⁢

g

∥

1

=

∥

f

∥

1

.

So, we have shown that

K

=

∥

f

∥

p

when

p

<

∞

and

∥

f

∥

p

<

∞

, and when

p

=

1

. From now on, it is assumed that the

measure

is

σ

-finite. Then there is a sequence

A

n

∈

𝔐

increasing

to the whole of

X

and such that

μ

⁢

(

A

n

)

<

∞

.

Now consider the case where

1

<

p

<

∞

and

∥

f

∥

p

=

∞

. Let

f

n

be the sequence of functions

f

n

=

1

A

n

⁢

1

|

f

|

≤

n

⁢

f

then,

|

f

n

|

≤

|

f

|

and

monotone

convergence gives

∥

f

n

∥

p

→

∥

f

∥

p

=

∞

. Therefore,

K

≥

sup

⁡

{

∥

f

n

⁢

g

∥

1

:

g

∈

L

q

,

∥

g

∥

q

=

1

}

=

∥

f

n

∥

p

.

and letting

n

go to

infinity

gives

K

=

∞

.

We finally consider

p

=

∞

. Then, for any

L

<

∥

f

∥

p

there exists a set

A

∈

𝔐

with

μ

⁢

(

A

)

>

0

such that

|

f

|

≥

L

on

A

.
Also, monotone convergence gives

μ

⁢

(

A

∩

A

n

)

→

μ

⁢

(

A

)

and, therefore,

μ

⁢

(

A

∩

A

n

)

>

0

eventually. Replacing

A

by

A

∩

A

n

if necessary, we may suppose that

μ

⁢

(

A

)

<

∞

. So, setting

g

=

1

A

/

μ

⁢

(

A

)

gives

∥

g

∥

1

=

1

and,

K

≥

∥

f

⁢

g

∥

1

=

∫

A

|

f

|

⁢

𝑑

μ

/

μ

⁢

(

A

)

≥

L

.

Letting

L

increase to

∥

f

∥

p

gives

K

≥

∥

f

∥

p

as required.

Title

proof of

L

p

-norm is dual to

L

q

Canonical name

ProofOfLpnormIsDualToLq

Date of creation

2013-03-22 18:38:16

Last modified on

2013-03-22 18:38:16

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 46E30

Classification

msc 28A25

Generated on Fri Feb 9 21:40:04 2018 by

LaTeXML
proof of Young Inequality

proof of Young Inequality

By the concavity of the

log

function

we have

log

⁡

a

⁢

b

=

1

p

⁢

log

⁡

a

p

+

1

q

⁢

log

⁡

b

q

≤

log

⁡

(

1

p

⁢

a

p

+

1

q

⁢

b

q

)

.

By

exponentiation

we obtain the desired result.

Title

proof of Young Inequality

Canonical name

ProofOfYoungInequality

Date of creation

2013-03-22 13:30:10

Last modified on

2013-03-22 13:30:10

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Proof

Classification

msc 46E30

Generated on Fri Feb 9 21:40:06 2018 by

LaTeXML
Young inequality

Young inequality

Let

a

,

b

>

0

and

p

,

q

∈

]

0

,

∞

[

with

1

/

p

+

1

/

q

=

1

. Then

a

⁢

b

≤

a

p

p

+

b

q

q

.

Title

Young inequality

Canonical name

YoungInequality

Date of creation

2013-03-22 13:30:08

Last modified on

2013-03-22 13:30:08

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

6

Author

paolini (1187)

Entry type

Theorem

Classification

msc 46E30

Related topic

YoungsInequality

Generated on Fri Feb 9 21:40:08 2018 by

LaTeXML
ℓ^p⁢(X) space

ℓ

p

⁢

(

X

)

space

\PMlinkescapephrase

property

0.0.1

Definition of

ℓ

p

⁢

(

X

)

Let

p

be a real number such that

1

≤

p

<

∞

.

Let

X

be a set and let

μ

be the counting measure on

X

, defined on the

σ

-

algebra

(

http://planetmath.org/SigmaAlgebra

)

𝔅

of all subsets of

X

. The

ℓ

p

⁢

(

X

)

space is a particular of a

L

p

-space (

http://planetmath.org/LpSpace

), defined as

ℓ

p

⁢

(

X

)

:=

L

p

⁢

(

X

,

𝔅

,

μ

)

Thus, the

ℓ

p

⁢

(

X

)

space consists of all

functions

f

:

X

⟶

ℂ

such that

∑

x

∈

X

|

f

⁢

(

x

)

|

p

<

∞

Of course, for the above sum to be finite one must necessarily have

f

⁢

(

x

)

≠

0

only for a

countable

number of

x

∈

X

(see this entry (

http://planetmath.org/SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable

)).

0.0.2

Properties

•

By the corresponding property on

L

p

-spaces, the space

ℓ

p

⁢

(

X

)

is a

Banach space

and its norm amounts to

∥

f

∥

p

=

(

∑

x

∈

X

|

f

⁢

(

x

)

|

p

)

1

p

•

By the corresponding property on

L

2

-spaces (

http://planetmath.org/L2SpacesAreHilbertSpaces

), the space

ℓ

2

⁢

(

X

)

is a

Hilbert space

and its

inner product

amounts to

⟨

f

,

g

⟩

=

∑

x

∈

X

f

⁢

(

x

)

⁢

g

⁢

(

x

)

¯

0.0.3

Nonseparability of

ℓ

p

⁢

(

X

)

for uncountable

X

- The space

ℓ

p

⁢

(

X

)

is

separable

if and only if

X

is a

countable set

. Moreover,

ℓ

p

⁢

(

X

)

admits a

Schauder basis

if and only if

X

is countable.

A Schauder basis for

ℓ

p

⁢

(

X

)

, when it exists, can be just the set of functions

{

δ

x

0

:

x

0

∈

X

}

defined by

δ

x

0

⁢

(

x

)

:=

{

1

,

if

⁢

x

=

x

0

0

if

⁢

x

≠

x

0

0.0.4

Orthonormal basis of

ℓ

2

⁢

(

X

)

The set of functions

{

δ

x

0

:

x

0

∈

X

}

is an

orthonormal basis

of

ℓ

2

⁢

(

X

)

. Hence, the

dimension

(

http://planetmath.org/OrthonormalBasis

) of

ℓ

2

⁢

(

X

)

is given by the

cardinality

of

X

(as all orthonormal bases have the same cardinality).

It can be shown that all Hilbert spaces are

isometrically isomorphic

(hence, preserving the inner product) to a

ℓ

2

⁢

(

X

)

space, for a suitable set

X

.

Title

ℓ

p

⁢

(

X

)

space

Canonical

name

ellpXSpace

Date of creation

2013-03-22 17:55:59

Last modified on

2013-03-22 17:55:59

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46E30

Classification

msc 46B26

Classification

msc 28B15

Synonym

ℓ

p

⁢

(

X

)

Synonym

ℓ

p

⁢

(

X

)

-space

Related topic

Lp

Related topic

ClassificationOfHilbertSpaces

Related topic

RieszFischerTheorem

Defines

ℓ

2

⁢

(

X

)

Defines

ℓ

2

⁢

(

X

)

space

Defines

ℓ

p

⁢

(

X

)

is nonseparable iff

X

is

uncountable

Defines

orthonormal basis of

ℓ

2

⁢

(

X

)

have the cardinality of

X

Generated on Fri Feb 9 21:40:11 2018 by

LaTeXML
Sobolev inequality

Sobolev inequality

For

1

≤

p

<

n

, define the

Sobolev conjugate

of

p

as

p

*

:=

n

⁢

p

n

-

p

.

Note that

-

n

/

p

*

=

1

-

n

/

p

.

In the following statement

∇

represent the

weak derivative

and

W

1

,

p

⁢

(

Ω

)

is the

Sobolev space

of

functions

u

∈

L

p

⁢

(

Ω

)

whose weak derivative

∇

⁡

u

is itself in

L

p

⁢

(

Ω

)

.

Theorem 1

Assume that

p

∈

[

1

,

n

)

and let

Ω

be a

bounded

,

open subset

of

R

n

with

Lipschitz

boundary

.
Then there is a

constant

C

>

0

such that, for all

u

∈

W

1

,

p

⁢

(

Ω

)

one has

∥

u

∥

L

p

*

⁢

(

Ω

)

≤

C

⁢

∥

∇

⁡

u

∥

L

p

⁢

(

Ω

)

.

We can restate the previous

Theorem

by saying that the Sobolev space

W

1

,

p

⁢

(

Ω

)

is a

subspace

of the Lebesgue space

L

p

*

⁢

(

Ω

)

and that the

inclusion map

i

:

W

1

,

p

⁢

(

Ω

)

→

L

q

*

⁢

(

Ω

)

is

continuous

.

Title

Sobolev inequality

Canonical name

SobolevInequality

Date of creation

2013-03-22 15:05:14

Last modified on

2013-03-22 15:05:14

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

10

Author

paolini (1187)

Entry type

Theorem

Classification

msc 46E35

Synonym

Sobolev embedding

Synonym

sobolev immersion

Synonym

Gagliardo Nirenberg inequality

Related topic

LpSpace

Defines

Sobolev conjugate

Defines

Sobolev exponent

Generated on Fri Feb 9 21:40:17 2018 by

LaTeXML