Quasi-analytic function explained

In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval [''a'',''b''] ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of [''a'',''b'']. Quasi-analytic classes are broader classes of functions for which this statement still holds true.

Definitions

Let

M=\{M_k\}

	infty

	k=0

be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions C^M([''a'',''b'']) is defined to be those f ∈ C^∞([''a'',''b'']) which satisfy

\left|

	d^kf
	dx^k

(x)\right|\leqA^k+1k!M_k

for all x ∈ [''a'',''b''], some constant A, and all non-negative integers k. If M_k = 1 this is exactly the class of real analytic functions on [''a'',''b''].

The class C^M([''a'',''b'']) is said to be quasi-analytic if whenever f ∈ C^M([''a'',''b'']) and

	d^kf
	dx^k

(x)=0

for some point x ∈ [''a'',''b''] and all k, then f is identically equal to zero.

A function f is called a quasi-analytic function if f is in some quasi-analytic class.

Quasi-analytic functions of several variables

For a function

f:R^n\toR

and multi-indexes

j=(j_1,j_2,\ldots,j

	n

	n)\inN

, denote

|j|=j_1+j_2+\ldots+j_n

, and

\partial^j

\partial

	j₁
x
	1

\partial

	j₂
x
	2

\ldots\partial

	j_n
x
	n

j!=j_1!j_2!\ldotsj_n!

and

	j₁
x
	1

	j₂
x
	2

\ldots

	j_n
x
	n

Then

is called quasi-analytic on the open set

U\subsetRⁿ

if for every compact

K\subsetU

there is a constant

such that

\left|D^{jf(x)\right|\leq}A^|j|+1j!M_|j|

for all multi-indexes

j\inNⁿ

and all points

x\inK

The Denjoy-Carleman class of functions of

variables with respect to the sequence

on the set

can be denoted

	M(U)
C
	n

, although other notations abound.

The Denjoy-Carleman class

	M(U)
C
	n

is said to be quasi-analytic when the only function in it having all its partial derivatives equal to zero at a point is the function identically equal to zero.

A function of several variables is said to be quasi-analytic when it belongs to a quasi-analytic Denjoy-Carleman class.

Quasi-analytic classes with respect to logarithmically convex sequences

In the definitions above it is possible to assume that

M₁₌₁

and that the sequence

M_k

is non-decreasing.

The sequence

M_k

is said to be logarithmically convex, if

M_k+1/M_k

is increasing.

When

M_k

is logarithmically convex, then

	1/k
(M
	k)

is increasing and

M_rM_s\leqM_r+s

for all

(r,s)\inN²

The quasi-analytic class

	M
C
	n

with respect to a logarithmically convex sequence

satisfies:

	M
C
	n

is a ring. In particular it is closed under multiplication.

	M
C
	n

is closed under composition. Specifically, if

f=(f_1,f_2,\ldotsf_p)\in

	M)
(C
	n

and

g\in

	M
C
	p

, then

g\circf\in

	M
C
	n

The Denjoy–Carleman theorem

The Denjoy–Carleman theorem, proved by after gave some partial results, gives criteria on the sequence M under which C^M([''a'',''b'']) is a quasi-analytic class. It states that the following conditions are equivalent:

C^M([''a'',''b'']) is quasi-analytic.

\sum1/L_j=infty

where

L_j=inf_k\ge(k ⋅

	1/k
M
	k

)

\sum_j

	1
	j

	*)
(M
	j

^-1/j=infty

, where M_j^* is the largest log convex sequence bounded above by M_j.

\sum

	*
M
	j-1

	*
(j+1)M
	j

=infty.

The proof that the last two conditions are equivalent to the second uses Carleman's inequality.

Example: pointed out that if M_n is given by one of the sequences

1,{(lnn)}^n,{(lnn)}^n{(lnlnn)}^n,{(lnn)}^n{(lnlnn)}^n{(lnlnlnn)}^n,...,

then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

Additional properties

For a logarithmically convex sequence

the following properties of the corresponding class of functions hold:

C^M

contains the analytic functions, and it is equal to it if and only if

\sup_j\geq

	1/j
(M
	j)

<infty

is another logarithmically convex sequence, with

M_j\leqC^jN_j

for some constant

, then

C^M\subsetC^N

C^M

is stable under differentiation if and only if

\sup_j\geq(M_j+1

	1/j
/M
	j)

<infty

For any infinitely differentiable function

there are quasi-analytic rings

C^M

and

C^N

and elements

g\inC^M

, and

h\inC^N

, such that

f=g+h

Weierstrass division

A function

g:R^n\toR

is said to be regular of order
d

with respect to
x_n

if

g(0,x_n)=h(x_n)x

	d

	n

and

h(0) ≠ 0

. Given

regular of order

with respect to

x_n

, a ring

A_n

of real or complex functions of

variables is said to satisfy the Weierstrass division with respect to
g

if for every

f\inA_n

there is

q\inA

, and

h_1,h_2,\ldots,h_d-1\inA_n-1

such that

f=gq+h

with

h(x',x_n)=\sum

	d-1

	j=0

h_j

	j
(x')x
	n

While the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property, the same is not true for other quasi-analytic classes.

is logarithmically convex and

C^M

is not equal to the class of analytic function, then

C^M

doesn't satisfy the Weierstrass division property with respect to

g(x_1,x_2,\ldots,x_n)=x_1+x

	2

	2