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Understanding the Convolution and smoothing

0

$begingroup$

here my question is what is mean by $f^epsilon:=eta_epsilon*f$ in $U_epsilon$

and how can we change form $U$ to $B(0,epsilon)$

in the molification definition and what is use convolution in sobolev spaces

and how can we prove that $int eta(x),dx = 1$

is that convulutions is well defined

sobolev-spaces

share|cite|improve this question

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

$endgroup$

add a comment | 

0

$begingroup$

here my question is what is mean by $f^epsilon:=eta_epsilon*f$ in $U_epsilon$

and how can we change form $U$ to $B(0,epsilon)$

in the molification definition and what is use convolution in sobolev spaces

and how can we prove that $int eta(x),dx = 1$

is that convulutions is well defined

sobolev-spaces

share|cite|improve this question

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

$endgroup$

add a comment | 

$begingroup$

here my question is what is mean by $f^epsilon:=eta_epsilon*f$ in $U_epsilon$

and how can we change form $U$ to $B(0,epsilon)$

in the molification definition and what is use convolution in sobolev spaces

and how can we prove that $int eta(x),dx = 1$

is that convulutions is well defined

sobolev-spaces

share|cite|improve this question

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

$endgroup$

share|cite|improve this question

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

share|cite|improve this question

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

edited Apr 2 at 19:35

Martín-Blas Pérez Pinilla

35.6k42972

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

asked Apr 2 at 15:53

Inverse ProblemInverse Problem

1,037919

1 Answer
1

active

oldest

votes

1

$begingroup$

(1) The sign $:=$ means definition.

(2) In the left integral $f(y)$ only makes sense for $yin U$. In the right integral, $eta_epsilon(y) = 0$ when $ynotin B(0,epsilon)$. For the equality see Proving commutativity of convolution $(f ast g)(x) = (g ast f)(x)$.

(3) What means "in the molification definition and what is use convolution in sobolev spaces"?

(4) We choose the constant $C$ for making $int_Bbb R^neta = 1$.

(5) As $f$ is locally integrable and $eta_epsilon$ has compact support the integrand (product of...) is integrable (in both integrals).

share|cite|improve this answer

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes where we use convulation
  $endgroup$
  – Inverse Problem
  Apr 3 at 2:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ..how can we say that $eta _epsilon in C^inffinty$
  $endgroup$
  – Inverse Problem
  Apr 3 at 5:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @InverseProblem, see andromeda.rutgers.edu/~loftin/ra1fal10/mollifier.pdf.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Apr 3 at 6:10
  

  
  
  
  

add a comment | 

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1

$begingroup$

(1) The sign $:=$ means definition.

(2) In the left integral $f(y)$ only makes sense for $yin U$. In the right integral, $eta_epsilon(y) = 0$ when $ynotin B(0,epsilon)$. For the equality see Proving commutativity of convolution $(f ast g)(x) = (g ast f)(x)$.

(3) What means "in the molification definition and what is use convolution in sobolev spaces"?

(4) We choose the constant $C$ for making $int_Bbb R^neta = 1$.

(5) As $f$ is locally integrable and $eta_epsilon$ has compact support the integrand (product of...) is integrable (in both integrals).

share|cite|improve this answer

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes where we use convulation
  $endgroup$
  – Inverse Problem
  Apr 3 at 2:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ..how can we say that $eta _epsilon in C^inffinty$
  $endgroup$
  – Inverse Problem
  Apr 3 at 5:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @InverseProblem, see andromeda.rutgers.edu/~loftin/ra1fal10/mollifier.pdf.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Apr 3 at 6:10
  

  
  
  
  

add a comment | 

1

$begingroup$

(1) The sign $:=$ means definition.

(2) In the left integral $f(y)$ only makes sense for $yin U$. In the right integral, $eta_epsilon(y) = 0$ when $ynotin B(0,epsilon)$. For the equality see Proving commutativity of convolution $(f ast g)(x) = (g ast f)(x)$.

(3) What means "in the molification definition and what is use convolution in sobolev spaces"?

(4) We choose the constant $C$ for making $int_Bbb R^neta = 1$.

(5) As $f$ is locally integrable and $eta_epsilon$ has compact support the integrand (product of...) is integrable (in both integrals).

share|cite|improve this answer

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes where we use convulation
  $endgroup$
  – Inverse Problem
  Apr 3 at 2:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ..how can we say that $eta _epsilon in C^inffinty$
  $endgroup$
  – Inverse Problem
  Apr 3 at 5:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @InverseProblem, see andromeda.rutgers.edu/~loftin/ra1fal10/mollifier.pdf.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Apr 3 at 6:10
  

  
  
  
  

add a comment | 

$begingroup$

(1) The sign $:=$ means definition.

(2) In the left integral $f(y)$ only makes sense for $yin U$. In the right integral, $eta_epsilon(y) = 0$ when $ynotin B(0,epsilon)$. For the equality see Proving commutativity of convolution $(f ast g)(x) = (g ast f)(x)$.

(3) What means "in the molification definition and what is use convolution in sobolev spaces"?

(4) We choose the constant $C$ for making $int_Bbb R^neta = 1$.

(5) As $f$ is locally integrable and $eta_epsilon$ has compact support the integrand (product of...) is integrable (in both integrals).

share|cite|improve this answer

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

$endgroup$

share|cite|improve this answer

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972

answered Apr 2 at 19:34

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

35.6k42972