Prev: NIKON D300
Next: proof by induction F_n >= 2^.5n for F_n fibonacci term n>=6
From: Joubert on 20 Jun 2010 09:06

> and with the pointwise
> multiplication you get an IR algebra. But I doubt that's your answer.

This looks promising

> Who did the notes? Why are you interested in them? You could try to
> post the outline of the topics and hope that someone can give you a clue
> for a good book about them.

Definition of the operator (Identity - Laplacian)^z
Definition of H^s via Fourier transform
Embeddings into C_0 in case s>n/2 and
counterexample in the case s=n/2
The trace operator from H^s to H^(s-1/2)
Algebra property in case s>n/2
Interpolation between Sobolev spaces
Paley Wiener theorem
Paley Wiener Schwartz theorem
General young inequality convolution between S'(R^n) and S(R^n)
Bernstein inequalities
Smooth dyadic partitions of unity in the Fourier variable
L.P. decomposition of a tempered distribution
Characterization of Sobolev and Holder spaces by L.P. decompos.
The Zygmund class.
Introduction to Besov spaces.
From: Joubert on 20 Jun 2010 09:09

> Not unless the product of two H_s functions is again an H_s function.
> And if true, that would answer the question.

Looks promising, too.

From: Bastian Erdnuess on 20 Jun 2010 10:46
A N Niel wrote:

> In article <slrni1r0h1.2nm.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>,
> Bastian Erdnuess <earthnut(a)web.de> wrote:
>
>> Joubert wrote:
>>
>> >> How would this make H_s to an algebra?
>> >
>> > Good question. Is H_s an algebra in any way?
>>
>> Sure, in many ways. H_s is e.g. a IR vector space with the pointwise
>> addition (modulus equivalence classes on H_s) and with the pointwise
>> multiplication you get an IR algebra.
>
> Not unless the product of two H_s functions is again an H_s function.
> And if true, that would answer the question.

Doesn't that follow by Cauchy-Schwarz already?

Cheers
Bastian
From: Rob Johnson on 24 Jun 2010 07:42
In article <slrni1saev.46s.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>,
Bastian Erdnuess <earthnut(a)we wrote:
>A N Niel wrote:
>
>> In article <slrni1r0h1.2nm.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>,
>> Bastian Erdnuess <earthnut(a)web.de> wrote:
>>
>>> Joubert wrote:
>>>
>>> >> How would this make H_s to an algebra?
>>> >
>>> > Good question. Is H_s an algebra in any way?
>>>
>>> Sure, in many ways. H_s is e.g. a IR vector space with the pointwise
>>> addition (modulus equivalence classes on H_s) and with the pointwise
>>> multiplication you get an IR algebra.
>>
>> Not unless the product of two H_s functions is again an H_s function.
>> And if true, that would answer the question.
>
>Doesn't that follow by Cauchy-Schwarz already?

Cauchy-Schwarz says that the product of two L^2 functions is an L^1
function. I don't see how that helps us here.

However, using the inequality

(1+|x|^2)^{s/2} <= 2^{s-1} ((1+|x-y|^2)^{s/2} + (1+|y|^2)^{s/2})

we can prove that H^s is an algebra for s > n/2.

Let F and G be the Fourier Transforms of f and g. Since f and g are
in H^s, we have that both

F(x) (1+|x|^2)^{s/2}

and

G(x) (1+|x|^2)^{s/2}

are in L^2. That is, both Sobolev norms

||f|| = ||F(x) (1+|x|^2)^{s/2}||
H^s 2
and

||g|| = ||G(x) (1+|x|^2)^{s/2}||
H^s 2

are finite. When s > n/2, (1+|x|^2)^{-s/2} is in L^2. Therefore,
define

C = ||(1+|x|^2)^{-s/2}||
s 2

C_s is finite when s > n/2. Holder says that both

F(x) = (F(x) (1+|x|^2)^{s/2}) (1+|x|^2)^{-s/2}

and

G(x) = (G(x) (1+|x|^2)^{s/2}) (1+|x|^2)^{-s/2}

are in L^1 when s > n/2. That is,

||F|| <= ||F(x) (1+|x|^2)^{s/2}|| ||(1+|x|^2)^{-s/2}||
1 2 2
= C ||f||
s H^s
and

||G|| <= ||G(x) (1+|x|^2)^{s/2}|| ||(1+|x|^2)^{-s/2}||
1 2 2
= C ||g||
s H^s

Notice that the Convolution Theorem says that

||fg|| = ||F*G(x) (1+|x|^2)^{s/2}||
H^s 2

which is the supremum of

|\ |\
| | F(x-y) G(y) H(x) (1+|x|^2)^{s/2} dy dx
\| \|

for all H in L^2 such that ||H||_2 <= 1.

|\ |\
| | F(x-y) G(y) H(x) (1+|x|^2)^{s/2} dy dx
\| \|

s-1 |\ |\
<= 2 | | F(x-y) (1+|x-y|^2)^{s/2} G(y) H(x) dy dx
\| \|

s-1 |\ |\
+ 2 | | F(x-y) G(y) (1+|y|^2)^{s/2} H(x) dy dx
\| \|

s-1 s-1
<= 2 ||f|| ||G|| ||H|| + 2 ||F|| ||g|| ||H||
H^s 1 2 1 H^s 2
s
<= 2 C ||f|| ||g|| ||H||
s H^s H^s 2

Thus,

s
||fg|| <= 2 C ||f|| ||g||
H^s s H^s H^s

where

C = ||(1+|x|^2)^{-s/2}||
s 2

Which is finite for s > n/2.

Let us compute C_s. Using polar coordinates in R^n, we get

2 |\oo 2 -s n-1
C = | (1 + r ) w r dr
s \| 0 n-1

where w_{n-1} is the "surface area" of the unit sphere in R^n. In
<http://www.whim.org/nebula/math/sphervol.html>, this is computed as

2 pi^{n/2}
w = ----------
n-1 Gamma(n/2)

Thus, if we let r^2 = t/(1-t), we get

2 2 pi^{n/2} |\oo 2 -s n-1
C = ---------- | (1 + r ) r dr
s Gamma(n/2) \| 0

pi^{n/2} |\1 s-n/2-1 n/2-1
= ---------- | (1 - t) t dt
Gamma(n/2) \| 0

pi^{n/2} Gamma(s-n/2) Gamma(n/2)
= ---------- -----------------------
Gamma(n/2) Gamma(s)

n/2 Gamma(s-n/2)
= pi ------------
Gamma(s)

Rob Johnson <rob(a)trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
From: Bastian Erdnuess on 24 Jun 2010 18:19
Rob Johnson wrote:

> In article <slrni1saev.46s.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>,
> Bastian Erdnuess <earthnut(a)we wrote:
>>A N Niel wrote:
>>
>>> In article <slrni1r0h1.2nm.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>,
>>> Bastian Erdnuess <earthnut(a)web.de> wrote:
>>>
>>>> Joubert wrote:
>>>>
>>>> >> How would this make H_s to an algebra?
>>>> >
>>>> > Good question. Is H_s an algebra in any way?
>>>>
>>>> Sure, in many ways. H_s is e.g. a IR vector space with the pointwise
>>>> addition (modulus equivalence classes on H_s) and with the pointwise
>>>> multiplication you get an IR algebra.
>>>
>>> Not unless the product of two H_s functions is again an H_s function.
>>> And if true, that would answer the question.
>>
>>Doesn't that follow by Cauchy-Schwarz already?
>
> Cauchy-Schwarz says that the product of two L^2 functions is an L^1
> function. I don't see how that helps us here.

That's right. There was a square on my scrap paper on a place where it
was not supposed to be.

But to emphasise my "in many ways" I'll give another way to turn H_s
into an algebra.

For s > n/2 each class in H_s contains a continuous function by Sobolev.
An element of H_s refers to this representant.

h(x) = exp(-|x|^2) is an element of H_s (I hope). For f,g e H_s setting
(f . g)(x) = (f(0) * g(0)) * exp(-|x|^2) turns H_s into a (boring,
non-unitary) algebra (I think).

Cheers
Bastian
First  |  Prev  |  Next  |  Last
Pages: 1 2 3
Prev: NIKON D300
Next: proof by induction F_n >= 2^.5n for F_n fibonacci term n>=6