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.

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

${\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq A^{k+1}k!M_{k}}${\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 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

${\displaystyle {\frac {d^{k}f}{dx^{k}}}(x)=0}${\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.

For a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } and multi-indexes ${\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}${\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}, denote ${\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}${\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}, and

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

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

and

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

Then ${\displaystyle f}${\displaystyle f} is called quasi-analytic on the open set ${\displaystyle U\subset \mathbb {R} ^{n}}${\displaystyle U\subset \mathbb {R} ^{n}} if for every compact ${\displaystyle K\subset U}${\displaystyle K\subset U} there is a constant ${\displaystyle A}${\displaystyle A} such that

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

for all multi-indexes ${\displaystyle j\in \mathbb {N} ^{n}}${\displaystyle j\in \mathbb {N} ^{n}} and all points ${\displaystyle x\in K}${\displaystyle x\in K}.

The Denjoy-Carleman class of functions of ${\displaystyle n}${\displaystyle n} variables with respect to the sequence ${\displaystyle M}${\displaystyle M} on the set ${\displaystyle U}${\displaystyle U} can be denoted ${\displaystyle C_{n}^{M}(U)}${\displaystyle C_{n}^{M}(U)}, although other notations abound.

The Denjoy-Carleman class ${\displaystyle C_{n}^{M}(U)}${\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.

In the definitions above it is possible to assume that ${\displaystyle M_{1}=1}${\displaystyle M_{1}=1} and that the sequence ${\displaystyle M_{k}}${\displaystyle M_{k}} is non-decreasing.

The sequence ${\displaystyle M_{k}}${\displaystyle M_{k}} is said to be logarithmically convex, if

${\displaystyle M_{k+1}/M_{k}}${\displaystyle M_{k+1}/M_{k}} is increasing.

When ${\displaystyle M_{k}}${\displaystyle M_{k}} is logarithmically convex, then ${\displaystyle (M_{k})^{1/k}}${\displaystyle (M_{k})^{1/k}} is increasing and

${\displaystyle M_{r}M_{s}\leq M_{r+s}}${\displaystyle M_{r}M_{s}\leq M_{r+s}} for all ${\displaystyle (r,s)\in \mathbb {N} ^{2}}${\displaystyle (r,s)\in \mathbb {N} ^{2}}.

The quasi-analytic class ${\displaystyle C_{n}^{M}}${\displaystyle C_{n}^{M}} with respect to a logarithmically convex sequence ${\displaystyle M}${\displaystyle M} satisfies:

• ${\displaystyle C_{n}^{M}}${\displaystyle C_{n}^{M}} is a ring. In particular it is closed under multiplication.
• ${\displaystyle C_{n}^{M}}${\displaystyle C_{n}^{M}} is closed under composition. Specifically, if ${\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}}${\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}} and ${\displaystyle g\in C_{p}^{M}}${\displaystyle g\in C_{p}^{M}}, then ${\displaystyle g\circ f\in C_{n}^{M}}${\displaystyle g\circ f\in C_{n}^{M}}.

The Denjoy–Carleman theorem, proved by Carleman (1926) after Denjoy (1921) 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.
• ${\displaystyle \sum 1/L_{j}=\infty }${\displaystyle \sum 1/L_{j}=\infty } where ${\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}${\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}.
• ${\displaystyle \sum _{j}{\frac {1}{j}}(M_{j}^{*})^{-1/j}=\infty }${\displaystyle \sum _{j}{\frac {1}{j}}(M_{j}^{*})^{-1/j}=\infty }, where M_j^* is the largest log convex sequence bounded above by M_j.
• ${\displaystyle \sum _{j}{\frac {M_{j-1}^{*}}{(j+1)M_{j}^{*}}}=\infty .}${\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 M_n is given by one of the sequences

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

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

For a logarithmically convex sequence ${\displaystyle M}${\displaystyle M} the following properties of the corresponding class of functions hold:

• ${\displaystyle C^{M}}${\displaystyle C^{M}} contains the analytic functions, and it is equal to it if and only if ${\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }${\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }
• If ${\displaystyle N}${\displaystyle N} is another logarithmically convex sequence, with ${\displaystyle M_{j}\leq C^{j}N_{j}}${\displaystyle M_{j}\leq C^{j}N_{j}} for some constant ${\displaystyle C}${\displaystyle C}, then ${\displaystyle C^{M}\subset C^{N}}${\displaystyle C^{M}\subset C^{N}}.
• ${\displaystyle C^{M}}${\displaystyle C^{M}} is stable under differentiation if and only if ${\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }${\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }.
• For any infinitely differentiable function ${\displaystyle f}${\displaystyle f} there are quasi-analytic rings ${\displaystyle C^{M}}${\displaystyle C^{M}} and ${\displaystyle C^{N}}${\displaystyle C^{N}} and elements ${\displaystyle g\in C^{M}}${\displaystyle g\in C^{M}}, and ${\displaystyle h\in C^{N}}${\displaystyle h\in C^{N}}, such that ${\displaystyle f=g+h}${\displaystyle f=g+h}.

A function ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} } is said to be regular of order ${\displaystyle d}${\displaystyle d} with respect to ${\displaystyle x_{n}}${\displaystyle x_{n}} if ${\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}}${\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}} and ${\displaystyle h(0)\neq 0}${\displaystyle h(0)\neq 0}. Given ${\displaystyle g}${\displaystyle g} regular of order ${\displaystyle d}${\displaystyle d} with respect to ${\displaystyle x_{n}}${\displaystyle x_{n}}, a ring ${\displaystyle A_{n}}${\displaystyle A_{n}} of real or complex functions of ${\displaystyle n}${\displaystyle n} variables is said to satisfy the Weierstrass division with respect to ${\displaystyle g}${\displaystyle g} if for every ${\displaystyle f\in A_{n}}${\displaystyle f\in A_{n}} there is ${\displaystyle q\in A}${\displaystyle q\in A}, and ${\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}}${\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}} such that

${\displaystyle f=gq+h}${\displaystyle f=gq+h} with ${\displaystyle h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}}${\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 ${\displaystyle M}${\displaystyle M} is logarithmically convex and ${\displaystyle C^{M}}${\displaystyle C^{M}} is not equal to the class of analytic function, then ${\displaystyle C^{M}}${\displaystyle C^{M}} doesn't satisfy the Weierstrass division property with respect to ${\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}${\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

• Smooth functions