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\}} ^{inf ty} ^{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 | \leq A ^{k+1} k! 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 ^{k} f)/(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)\in N} , denote |j| = j _{1+j} _{2+\ldots+j} _{n} , and D ^{j = (\partial ^{j})/(\partial x ^{j _{1}} _{1} \partial x ^{j _{2}} _{2} \ldots\partial x ^{j _{n}} _{n})} j! = j _{1!j} _{2!\ldots} j _{n!} and x ^{j _{1}} _{1} x ^{j _{2}} _{2} \ldots x ^{j _{n}} _{n} .

Then

f is called quasi-analytic on the open set U\subset R ^{n} if for every compact K\subset U there is a constant A such that \left|D ^{jf(x)\right|\leq} A ^{|j|+1} j!M _{|j|}

for all multi-indexes

j\in N ^{n} and all points x\in K .

The Denjoy-Carleman class of functions of

n variables with respect to the sequence M on the set U can be denoted C ^{M(U)} _{n} , although other notations abound.

The Denjoy-Carleman class

C ^{M(U)} _{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

M _{1=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 (M ^{1 /k} _{k)} is increasing and M _{rM} _{s\leq} M _{r+s} for all (r,s)\in N ^{2} .

The quasi-analytic class

C ^{M} _{n} with respect to a logarithmically convex sequence M satisfies:C ^{M} _{n} is a ring. In particular it is closed under multiplication.C ^{M} _{n} is closed under composition. Specifically, if f = (f _{1,f} _{2,\ldots} f _{p)\in} (C ^{M)} _{n} ^{p} and g\in C ^{M} _{p} , then g\circ f\in C ^{M} _{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.

\sum 1 /L _{j} = inf ty where L _{j=} inf _{k\ge} (k ⋅ M ^{1 /k} _{k} ) .\sum _{j} (1)/(j) (M ^{*)} _{j} ^{-1/j} = inf ty , where M_j^* is the largest log convex sequence bounded above by M_j.\sum _{j (M ^{*} _{j -1})/((j +1 )M ^{*} _{j})} = inf ty.

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 , {( ln n)} ^{n,} {( ln n)} ^{n{(ln} ln n)} ^{n,} {( ln n)} ^{n{(ln} ln n)} ^{n{(ln} ln ln n)} ^{n,} ... , then the corresponding class is quasi-analytic. The first sequence gives analytic functions.

Additional properties

For a logarithmically convex sequence

M 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} (M ^{1 /j} _{j)} < inf ty

• If

N is another logarithmically convex sequence, with M _{j\leq} C ^{j} N _{j} for some constant C , then C ^{M\subset} C ^{N} .C ^{M} is stable under differentiation if and only if \sup _{j\geq} (M _{j+1} /M ^{1 /j} _{j)} < inf ty .

• For any infinitely differentiable function

f there are quasi-analytic rings C ^{M} and C ^{N} and elements g\in C ^{M} , and h\in C ^{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 g regular of order d with respect to x _{n} , a ring A _{n} of real or complex functions of n variables is said to satisfy the Weierstrass division with respect to g if for every f\in A _{n} there is q\in A , and h _{1,h} _{2,\ldots,h} _{d-1} \in A _{n-1} such that f = gq + h with h(x',x _{n)=\sum} ^{d -1} ^{j =0} h _{j} (x')x ^{j} _{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.

If

M 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}