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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}} .

• 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