Distributions

1) complete normato Spazio or of Banach:

Space C(K) of the continuous functions on an interval closed and limited containing K the origin possesses the norm ||f||= sup|f(x)| and in it every succession of Cauchy is convergent.

 

2) Works to delineate they continuous:

Draft of linear a T application and is continued that it associates to a function a real number, in symbols writes C(K) ' f ® .

 

3) Define the property of linearity and continuity for works to delineate them continuous:

to) Linearity :

b) Continuity : if f_j ® f in C(K) ž or analogous

if f_j ® 0 in C(K) ž

 

4) Example of works to delineate them continuous :

Its continuity is demonstrated in how much if f_j®0 has in how much .

 

5) Measure:

Draft of works them linear continues on C(K).

 

6)       Measure of Dirac:

It is works to delineate them continuous defined from the but it is not associated to some sommabile function therefore is necessary to characterize a succession of sommabili functions{j_n} that approximates it.

7) When a succession of functions approximates d :

A succession of functions {j_n} approximates the delta of Dirac if for every f ? C(K)

 

8) Support of one function:

He is the complementary one of the greatest open in which the function is null.

 

9) Function to compact support:

Draft of one null function ovunque with the exception of a limited interval.

10) Describe the D(space "):

It is the space of the functions to compact support infinite derivabili times on ".

 

11) Definition of convergence on D( "):

One succession of functions {f_j}®0 if :

to)    a compact interval K exists to outside of which f_j = 0 for every j except to more an ended number.

b)    every for n = 0,1,2...

 

12) Works to delineate they continuous on D( ") :

Draft of a T application that associates to one function f ? D( ") a real number, in symbols is written

D( ") ' f ® . The continuity of the T application is from agreeing in the sense that if f_j®0 with the definition of convergence on D( ") then These works is said them distributions.

 

13) Distribution :

Draft of works to delineate them continuous on D( "), a distribution example is the d of Dirac.

 

14) Describe the space Of ( ") :

Draft of the space formed from the distributions.

 

15) Derivative of one distribution:

It is defined from eguaglianza the with f ? D( ")

It is demonstrated simply integrating for parts in fact but the term to the inside of the parenthesis quadrant is null in how much jthe (t) is like dthe (t) and therefore it is worth 0 to the edges of the real axis.

The distribuzionali derivatives coincide with the classic derivatives in the continuity points while they are various in the points of discontinuità. is as an example succeeded to demonstrate that the derivative of the unitary step is the d of Dirac.

 

16) Describe the S(space ") :

It is the space of the infinitely derivabili to fast decrease, such functions that is that

for every n and m. it is contained to the inside of the D(space ") of the infinite functions derivabili times to compact support.

 

17) Describe the S' space ( ") :

It is the space of the distributions moderated that is of works them linear continues on S( "). An example of moderated distribution is the d of Dirac and more in a generalized manner it can be said that a t distribution is moderated if its product of convoluzione with j has one increase to polynomial .

 

18) relative Formula to Transformed of Laplace of one the distribution :