Quasi-analytic function

Quasi-analytic function   (Redirected from Denjoy–Carleman theorem) Jump to navigation Jump to search 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.

Contents 1 Definitions 1.1 Quasi-analytic functions of several variables 1.2 Quasi-analytic classes with respect to logarithmically convex sequences 2 The Denjoy–Carleman theorem 3 Additional properties 3.1 Weierstrass division 4 References Definitions Let {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 {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 {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 {displaystyle f:mathbb {R} ^{n}to mathbb {R} } and multi-indexes {displaystyle j=(j_{1},j_{2},ldots ,j_{n})in mathbb {N} ^{n}} , denote {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 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}}.} Then {displaystyle f} is called quasi-analytic on the open set {displaystyle Usubset mathbb {R} ^{n}} if for every compact {displaystyle Ksubset U} there is a constant {displaystyle A} such that {displaystyle left|D^{j}f(x)right|leq A^{|j|+1}j!M_{|j|}} for all multi-indexes {displaystyle jin mathbb {N} ^{n}} and all points {displaystyle xin K} .

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

The Denjoy-Carleman class {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 {displaystyle M_{1}=1} and that the sequence {displaystyle M_{k}} is non-decreasing.

The sequence {displaystyle M_{k}} is said to be logarithmically convex, if {displaystyle M_{k+1}/M_{k}} is increasing.

When {displaystyle M_{k}} is logarithmically convex, then {displaystyle (M_{k})^{1/k}} is increasing and {displaystyle M_{r}M_{s}leq M_{r+s}} for all {displaystyle (r,s)in mathbb {N} ^{2}} .

The quasi-analytic class {displaystyle C_{n}^{M}} with respect to a logarithmically convex sequence {displaystyle M} satisfies: {displaystyle C_{n}^{M}} is a ring. In particular it is closed under multiplication. {displaystyle C_{n}^{M}} is closed under composition. Specifically, if {displaystyle f=(f_{1},f_{2},ldots f_{p})in (C_{n}^{M})^{p}} and {displaystyle gin C_{p}^{M}} , then {displaystyle gcirc fin 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. {displaystyle sum 1/L_{j}=infty } where {displaystyle L_{j}=inf _{kgeq j}(kcdot M_{k}^{1/k})} . {displaystyle sum _{j}{frac {1}{j}}(M_{j}^{*})^{-1/j}=infty } , where Mj* is the largest log convex sequence bounded above by Mj. {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 {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 {displaystyle M} the following properties of the corresponding class of functions hold: {displaystyle C^{M}} contains the analytic functions, and it is equal to it if and only if {displaystyle sup _{jgeq 1}(M_{j})^{1/j}

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