Chapter 32

Fourier Transforms32.1 An Algebra Of Special Functions

First recall the following definition of a polynomial.

Definition 32.1.1 α = (α1, · · · ,αn) for α1 · · ·αn positive integers is called a multi-index.For α a multi-index, |α| ≡ α1 + · · ·+αn and if x ∈ Rn,

x = (x1, · · · ,xn) ,

and f a function, definexα ≡ xα1

1 xα22 · · ·x

αnn .

A polynomial in n variables of degree m is a function of the form

p(x) = ∑|α|≤m

aα xα .

Here α is a multi-index as just described and aα ∈ C. Also define for α = (α1, · · · ,αn) amulti-index

Dα f (x)≡ ∂ |α| f∂xα1

1 ∂xα22 · · ·∂xαn

n.

Definition 32.1.2 Define G1 to be the functions of the form p(x)e−a|x|2 where a > 0 andp(x) is a polynomial. Let G be all finite sums of functions in G1. Thus G is an algebra offunctions which has the property that if f ∈ G then f ∈ G .

It is always assumed, unless stated otherwise that the measure will be Lebesgue mea-sure.

Lemma 32.1.3 G is dense in C0 (Rn) with respect to the norm,

|| f ||∞≡ sup{| f (x)| : x ∈ Rn}

Proof: By the Weierstrass approximation theorem, it suffices to show G separates thepoints and annihilates no point. It was already observed in the above definition that f ∈ Gwhenever f ∈ G . If y1 ̸= y2 suppose first that |y1| ̸= |y2| . Then in this case, you can letf (x) ≡ e−|x|

2and f ∈ G and f (y1) ̸= f (y2). If |y1| = |y2| , then suppose y1k ̸= y2k. This

must happen for some k because y1 ̸= y2. Then let f (x) ≡ xke−|x|2. Thus G separates

points. Now e−|x|2

is never equal to zero and so G annihilates no point of Rn. This provesthe lemma.

These functions are clearly quite specialized. Therefore, the following theorem is some-what surprising.

Theorem 32.1.4 For each p≥ 1, p < ∞,G is dense in Lp (Rn).

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