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Distribution Theory/Bump functions

Preliminary definitionsEdit

Definition:

Let   be a function, where   is an open subset of  . We say

  •   iff all partial derivatives of   up to order   exist and are continuous
  •   iff all partial derivatives of   of any order exist and are continuous.

Definition:

Let   be a topological space and let   be a function. Then the support of   is defined to be the set

 ;

the bar above the set on the right denotes the topological closure.

Definition:

A bump function is a function   from an open set   to   such that the following two conditions are satisfied:

  1.   is compact
  2.  

Multiindex notationEdit

The multiindex notation is an efficient way of denoting several things in multi-dimensional space. For instance, it takes fairly long to denote a partial derivative in the usual way; in the usual notation, a partial derivative is denoted

 

for some  . Now in multiindex notation, the   are assembled into a vector  , and the term

 

is then used instead of the partial derivative notation used above. Now, for one partial derivative this may not be a huge advantage (unless one is talking about a general partial derivative), but for instance when one sums all partial derivatives of a polynomial  , say, then one obtains expressions as such:

  (Note that this is well-defined, as the sum is finite.)

Now compare this to the much longer

 ;

as you can see, we saved a lot of time, and that's what's all about. Multiindex notation was invented by Laurent Schwartz.

Other multiindex conventions are the following (we use a convention by Béla Bollobás and denote  ):

  • Multiindex Partial order:  
  • Multiindex factorial:  
  • Multiindex binomial coefficient: Let   and   be multiindices, then  
  • Multiindex power: Let additionally  , then set  
  • Constant multiindex: If  , we denote the constant multiindex   by the boldface  
  • Multiindex differentiability: We write   iff the partial derivatives   exist for all   with  .

Further, the absolute value of a multiindex   is defined as

 .

A few sample theorems on multiindices are these (we'll need them often):

Theorem (multiindex binomial formula):

Let   be a multiindex,  . Then

 .

Note that this formula looks exactly as in the one-dimensional case, with one dimensional variables replaced by multiindex variables. This will be a recurrent phenomenon.

Proof:

We prove the theorem by induction on  . For   the case is clear. Now suppose the theorem has been proven where  , and let instead  . Then   has at least one nonzero component; let's say the  -th component of   is nonzero. Then   (  denoting the  -th unit vector, i.e.  ) is a multiindex of absolute value  . By induction,

 

and hence, multiplying both sides by  ,

 

because

 

by the respective rule for the usual  -dim. binomial coefficient. 

Theorem (multiindex product rule):

Let   be a multiindex,   be open and  . Then

 ;

in particular,  .

Proof:

Again, we proceed by induction on  . As before, pick   such that the  -th entry of   is nonzero, and define  . Then by induction

  

Note that the proof is essentially the same as in the previous theorem, since by the product rule, differentiation in one direction has the same effect as multiplying the "sum of derivatives" to the existing derivatives.

Note that the dimension of the respective multiindex must always match the dimension of the space we are considering.

Stability properties, TVS of bump functions, convergenceEdit