• Source: Quasi-analytic function

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



}

k
=
0


∞




{\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}

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





|





d

k


f


d

x

k





(
x
)

|

≤

A

k
+
1


k
!

M

k




{\displaystyle \left|{\frac {d^{k}f}{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 Mk = 1 this is exactly the class of real analytic functions on [a,b].
The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and








d

k


f


d

x

k





(
x
)
=
0


{\displaystyle {\frac {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


→

R



{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }

and multi-indexes



j
=
(

j

1


,

j

2


,
…
,

j

n


)
∈


N


n




{\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}

, denote




|

j

|

=

j

1


+

j

2


+
…
+

j

n




{\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}

, and





D

j


=



∂

j



∂

x

1



j

1




∂

x

2



j

2




…
∂

x

n



j

n









{\displaystyle D^{j}={\frac {\partial ^{j}}{\partial x_{1}^{j_{1}}\partial x_{2}^{j_{2}}\ldots \partial x_{n}^{j_{n}}}}}





j
!
=

j

1


!

j

2


!
…

j

n


!


{\displaystyle j!=j_{1}!j_{2}!\ldots j_{n}!}


and





x

j


=

x

1



j

1





x

2



j

2




…

x

n



j

n




.


{\displaystyle x^{j}=x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{n}^{j_{n}}.}


Then



f


{\displaystyle f}

is called quasi-analytic on the open set



U
⊂


R


n




{\displaystyle U\subset \mathbb {R} ^{n}}

if for every compact



K
⊂
U


{\displaystyle K\subset U}

there is a constant



A


{\displaystyle A}

such that





|


D

j


f
(
x
)

|

≤

A


|

j

|

+
1


j
!

M


|

j

|





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


for all multi-indexes



j
∈


N


n




{\displaystyle j\in \mathbb {N} ^{n}}

and all points



x
∈
K


{\displaystyle x\in K}

.
The Denjoy-Carleman class of functions of



n


{\displaystyle n}

variables with respect to the sequence



M


{\displaystyle M}

on the set



U


{\displaystyle U}

can be denoted




C

n


M


(
U
)


{\displaystyle C_{n}^{M}(U)}

, although other notations abound.
The Denjoy-Carleman class




C

n


M


(
U
)


{\displaystyle C_{n}^{M}(U)}

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


{\displaystyle M_{1}=1}

and that the sequence




M

k




{\displaystyle M_{k}}

is non-decreasing.
The sequence




M

k




{\displaystyle M_{k}}

is said to be logarithmically convex, if





M

k
+
1



/


M

k




{\displaystyle M_{k+1}/M_{k}}

is increasing.
When




M

k




{\displaystyle M_{k}}

is logarithmically convex, then



(

M

k



)

1

/

k




{\displaystyle (M_{k})^{1/k}}

is increasing and





M

r



M

s


≤

M

r
+
s




{\displaystyle M_{r}M_{s}\leq M_{r+s}}

for all



(
r
,
s
)
∈


N


2




{\displaystyle (r,s)\in \mathbb {N} ^{2}}

.
The quasi-analytic class




C

n


M




{\displaystyle C_{n}^{M}}

with respect to a logarithmically convex sequence



M


{\displaystyle M}

satisfies:





C

n


M




{\displaystyle C_{n}^{M}}

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





C

n


M




{\displaystyle C_{n}^{M}}

is closed under composition. Specifically, if



f
=
(

f

1


,

f

2


,
…

f

p


)
∈
(

C

n


M



)

p




{\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}}

and



g
∈

C

p


M




{\displaystyle g\in C_{p}^{M}}

, then



g
∘
f
∈

C

n


M




{\displaystyle g\circ f\in C_{n}^{M}}

.




The Denjoy–Carleman theorem


The Denjoy–Carleman theorem, proved by Carleman (1926) after Denjoy (1921) gave some partial results, gives criteria on the sequence M under which CM([a,b]) is a quasi-analytic class. It states that the following conditions are equivalent:

CM([a,b]) is quasi-analytic.




∑
1

/


L

j


=
∞


{\displaystyle \sum 1/L_{j}=\infty }

where




L

j


=

inf

k
≥
j


(
k
⋅

M

k


1

/

k


)


{\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}

.





∑

j




1
j


(

M

j


∗



)

−
1

/

j


=
∞


{\displaystyle \sum _{j}{\frac {1}{j}}(M_{j}^{*})^{-1/j}=\infty }

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





∑

j





M

j
−
1


∗



(
j
+
1
)

M

j


∗





=
∞
.


{\displaystyle \sum _{j}{\frac {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: Denjoy (1921) pointed out that if Mn 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


,
…
,


{\displaystyle 1,\,{(\ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n},\,{(\ln n)}^{n}\,{(\ln \ln n)}^{n}\,{(\ln \ln \ln n)}^{n},\dots ,}


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




Additional properties


For a logarithmically convex sequence



M


{\displaystyle M}

the following properties of the corresponding class of functions hold:





C

M




{\displaystyle C^{M}}

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




sup

j
≥
1


(

M

j



)

1

/

j


<
∞


{\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }


If



N


{\displaystyle N}

is another logarithmically convex sequence, with




M

j


≤

C

j



N

j




{\displaystyle M_{j}\leq C^{j}N_{j}}

for some constant



C


{\displaystyle C}

, then




C

M


⊂

C

N




{\displaystyle C^{M}\subset C^{N}}

.





C

M




{\displaystyle C^{M}}

is stable under differentiation if and only if




sup

j
≥
1


(

M

j
+
1



/


M

j



)

1

/

j


<
∞


{\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }

.
For any infinitely differentiable function



f


{\displaystyle f}

there are quasi-analytic rings




C

M




{\displaystyle C^{M}}

and




C

N




{\displaystyle C^{N}}

and elements



g
∈

C

M




{\displaystyle g\in C^{M}}

, and



h
∈

C

N




{\displaystyle h\in C^{N}}

, such that



f
=
g
+
h


{\displaystyle f=g+h}

.




= Weierstrass division

=
A function



g
:


R


n


→

R



{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }

is said to be regular of order



d


{\displaystyle d}

with respect to




x

n




{\displaystyle x_{n}}

if



g
(
0
,

x

n


)
=
h
(

x

n


)

x

n


d




{\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}}

and



h
(
0
)
≠
0


{\displaystyle h(0)\neq 0}

. Given



g


{\displaystyle g}

regular of order



d


{\displaystyle d}

with respect to




x

n




{\displaystyle x_{n}}

, a ring




A

n




{\displaystyle A_{n}}

of real or complex functions of



n


{\displaystyle n}

variables is said to satisfy the Weierstrass division with respect to



g


{\displaystyle g}

if for every



f
∈

A

n




{\displaystyle f\in A_{n}}

there is



q
∈
A


{\displaystyle q\in A}

, and




h

1


,

h

2


,
…
,

h

d
−
1


∈

A

n
−
1




{\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}}

such that




f
=
g
q
+
h


{\displaystyle f=gq+h}

with



h
(

x
′

,

x

n


)
=

∑

j
=
0


d
−
1



h

j


(

x
′

)

x

n


j




{\displaystyle h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}}

.
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


{\displaystyle M}

is logarithmically convex and




C

M




{\displaystyle C^{M}}

is not equal to the class of analytic function, then




C

M




{\displaystyle C^{M}}

doesn't satisfy the Weierstrass division property with respect to



g
(

x

1


,

x

2


,
…
,

x

n


)
=

x

1


+

x

2


2




{\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}

.




References


Carleman, T. (1926), Les fonctions quasi-analytiques, Gauthier-Villars
Cohen, Paul J. (1968), "A simple proof of the Denjoy-Carleman theorem", The American Mathematical Monthly, 75 (1), Mathematical Association of America: 26–31, doi:10.2307/2315100, ISSN 0002-9890, JSTOR 2315100, MR 0225957
Denjoy, A. (1921), "Sur les fonctions quasi-analytiques de variable réelle", C. R. Acad. Sci. Paris, 173: 1329–1331
Hörmander, Lars (1990), The Analysis of Linear Partial Differential Operators I, Springer-Verlag, ISBN 3-540-00662-1
Leont'ev, A.F. (2001) [1994], "Quasi-analytic class", Encyclopedia of Mathematics, EMS Press
Solomentsev, E.D. (2001) [1994], "Carleman theorem", Encyclopedia of Mathematics, EMS Press

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