Distribution Theory/Bump functions

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Preliminary definitions[edit]

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

Multiindex notation[edit]

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, convergence[edit]