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Quasi-analytic function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, a quasi-analytic class of functions is a generalization of the class of real <a href="Analytic_function" title="Analytic function">analytic functions</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}</annotation>

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</math><img src="./e0c531edd4d521fc8634fd39c8603f6c8faaf3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.398ex; height:3.009ex;" alt="{\displaystyle M=\{M_{k}\}_{k=0}^{\infty }}" loading="lazy"> 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

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<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq A^{k+1}k!M_{k}}</annotation>

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</math><img src="./d9ab3d98f6ff04a7a1ae7f7bbe089fc5c2f6eba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.135ex; height:6.843ex;" alt="{\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq A^{k+1}k!M_{k}}" loading="lazy"></dd></dl>

for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = 1 this is exactly the class of real <a href="Analytic_function" title="Analytic function">analytic functions</a> on [a,b].

The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{k}f}{dx^{k}}}(x)=0}</annotation>

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</math><img src="./5c3afd586f644ee2d7d53bf14f35a8c8e8500da8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.87ex; height:6.176ex;" alt="{\displaystyle {\frac {d^{k}f}{dx^{k}}}(x)=0}" loading="lazy"></dd></dl>

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.

<div class="mw-heading mw-heading3"><h3 id="Quasi-analytic_functions_of_several_variables">Quasi-analytic functions of several variables</h3></div>

For a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

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</math><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> and multi-indexes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}</annotation>

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</math><img src="./2d36f573786c46362f5bdf7c4d22113948d04f22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:24.043ex; height:2.843ex;" alt="{\displaystyle j=(j_{1},j_{2},\ldots ,j_{n})\in \mathbb {N} ^{n}}" loading="lazy">, denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}</annotation>

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</math><img src="./bdc28f001ff880718527f1a94480add633f53e5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.796ex; height:2.843ex;" alt="{\displaystyle |j|=j_{1}+j_{2}+\ldots +j_{n}}" loading="lazy">, and

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<annotation encoding="application/x-tex">{\displaystyle D^{j}={\frac {\partial ^{j}}{\partial x_{1}^{j_{1}}\partial x_{2}^{j_{2}}\ldots \partial x_{n}^{j_{n}}}}}</annotation>

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</math><img src="./d4ebbfe2a7682976c35e89eadb1dd89547fca01b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.566ex; height:7.176ex;" alt="{\displaystyle D^{j}={\frac {\partial ^{j}}{\partial x_{1}^{j_{1}}\partial x_{2}^{j_{2}}\ldots \partial x_{n}^{j_{n}}}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle j!=j_{1}!j_{2}!\ldots j_{n}!}</annotation>

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</math><img src="./eafa0a9481bf36f454f255a1f0762c92c04d5b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:16.369ex; height:2.509ex;" alt="{\displaystyle j!=j_{1}!j_{2}!\ldots j_{n}!}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle x^{j}=x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{n}^{j_{n}}.}</annotation>

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</math><img src="./98e9f786604d4744ad383069e509bfa00a579c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.828ex; height:3.509ex;" alt="{\displaystyle x^{j}=x_{1}^{j_{1}}x_{2}^{j_{2}}\ldots x_{n}^{j_{n}}.}" loading="lazy"></dd></dl>

Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is called quasi-analytic on the open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {R} ^{n}}</annotation>

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</math><img src="./c1caefb347c86337ea7cd0c354acd2294bd7d81d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U\subset \mathbb {R} ^{n}}" loading="lazy"> if for every compact <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset U}">

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<annotation encoding="application/x-tex">{\displaystyle K\subset U}</annotation>

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</math><img src="./535ed682dca3a92174f9efae55ca7ddd073567c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.947ex; height:2.176ex;" alt="{\displaystyle K\subset U}" loading="lazy"> there is a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle \left|D^{j}f(x)\right|\leq A^{|j|+1}j!M_{|j|}}</annotation>

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</math><img src="./09cf92690422af1cf474470ce913462c6ce516c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.995ex; height:3.676ex;" alt="{\displaystyle \left|D^{j}f(x)\right|\leq A^{|j|+1}j!M_{|j|}}" loading="lazy"></dd></dl>

for all multi-indexes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in \mathbb {N} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle j\in \mathbb {N} ^{n}}</annotation>

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</math><img src="./33a287c40deb2a0115515f30ebe88ad36d15b60c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.722ex; height:2.676ex;" alt="{\displaystyle j\in \mathbb {N} ^{n}}" loading="lazy"> and all points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}">

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<annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation>

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</math><img src="./f9075ef2df706bf2764f76182a1ddf55ff231a92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}" loading="lazy">.

The Denjoy-Carleman class of functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables with respect to the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> can be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}^{M}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle C_{n}^{M}(U)}</annotation>

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</math><img src="./8287cd2e16537dd3ecdf05507bd2ac061d6c80b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.349ex; height:3.009ex;" alt="{\displaystyle C_{n}^{M}(U)}" loading="lazy">, although other notations abound.

The Denjoy-Carleman class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}^{M}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle C_{n}^{M}(U)}</annotation>

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</math><img src="./8287cd2e16537dd3ecdf05507bd2ac061d6c80b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.349ex; height:3.009ex;" alt="{\displaystyle C_{n}^{M}(U)}" loading="lazy"> 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.

<div class="mw-heading mw-heading3"><h3 id="Quasi-analytic_classes_with_respect_to_logarithmically_convex_sequences">Quasi-analytic classes with respect to logarithmically convex sequences</h3></div>

In the definitions above it is possible to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=1}">

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</math><img src="./d2921d2315ff20078a0c075f6614bea6b2debcda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.569ex; height:2.509ex;" alt="{\displaystyle M_{1}=1}" loading="lazy"> and that the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle M_{k}}</annotation>

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</math><img src="./43175c381ee6ba2694520af1f1a26c676a2726ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}" loading="lazy"> is non-decreasing.

The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}">

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</math><img src="./43175c381ee6ba2694520af1f1a26c676a2726ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}" loading="lazy"> is said to be logarithmically convex, if

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<annotation encoding="application/x-tex">{\displaystyle M_{k+1}/M_{k}}</annotation>

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</math><img src="./a4e855cdfc2a19a9b7f51334e13030463020e9f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.949ex; height:2.843ex;" alt="{\displaystyle M_{k+1}/M_{k}}" loading="lazy"> is increasing.</dd></dl>

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}">

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</math><img src="./43175c381ee6ba2694520af1f1a26c676a2726ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}" loading="lazy"> is logarithmically convex, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{k})^{1/k}}">

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<annotation encoding="application/x-tex">{\displaystyle (M_{k})^{1/k}}</annotation>

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</math><img src="./7963911cf1e088bed1dc4230f6df81d26ff666f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.885ex; height:3.343ex;" alt="{\displaystyle (M_{k})^{1/k}}" loading="lazy"> is increasing and

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<annotation encoding="application/x-tex">{\displaystyle M_{r}M_{s}\leq M_{r+s}}</annotation>

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</math><img src="./b601aa03d6fc360f948b23b461239243550311d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.861ex; height:2.509ex;" alt="{\displaystyle M_{r}M_{s}\leq M_{r+s}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,s)\in \mathbb {N} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle (r,s)\in \mathbb {N} ^{2}}</annotation>

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</math><img src="./8a09bf51a8e82f8b075799cfeda2bf4ef5e11e65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.555ex; height:3.176ex;" alt="{\displaystyle (r,s)\in \mathbb {N} ^{2}}" loading="lazy">.</dd></dl>

The quasi-analytic class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}^{M}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{n}^{M}}</annotation>

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</math><img src="./18becd4d6169c10395327a7fe3461f3a44822378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.757ex; height:2.843ex;" alt="{\displaystyle C_{n}^{M}}" loading="lazy"> with respect to a logarithmically convex sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> satisfies:

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<annotation encoding="application/x-tex">{\displaystyle C_{n}^{M}}</annotation>

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</math><img src="./18becd4d6169c10395327a7fe3461f3a44822378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.757ex; height:2.843ex;" alt="{\displaystyle C_{n}^{M}}" loading="lazy"> is a ring. In particular it is closed under multiplication.</li>

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<annotation encoding="application/x-tex">{\displaystyle C_{n}^{M}}</annotation>

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</math><img src="./18becd4d6169c10395327a7fe3461f3a44822378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.757ex; height:2.843ex;" alt="{\displaystyle C_{n}^{M}}" loading="lazy"> is closed under composition. Specifically, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}}">

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</mrow>

</msub>

<msub>

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</msub>

</mrow>

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</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}}</annotation>

</semantics>

</math><img src="./ccdcee5f37a0b68691cd8723ac5d8f96af4a29b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.416ex; height:3.176ex;" alt="{\displaystyle f=(f_{1},f_{2},\ldots f_{p})\in (C_{n}^{M})^{p}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in C_{p}^{M}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in C_{p}^{M}}</annotation>

</semantics>

</math><img src="./4ba2b8aed85ce7e4422b9fe486dded824c028750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.714ex; height:3.176ex;" alt="{\displaystyle g\in C_{p}^{M}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f\in C_{n}^{M}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f\in C_{n}^{M}}</annotation>

</semantics>

</math><img src="./9fd3325d4b1d3d103277cb0638e52edc480cfabf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.187ex; height:2.843ex;" alt="{\displaystyle g\circ f\in C_{n}^{M}}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="The_Denjoy–Carleman_theorem">The Denjoy–Carleman theorem</h2></div>

The Denjoy–Carleman theorem, proved by <a href="#CITEREFCarleman1926">Carleman (1926)</a> after <a href="#CITEREFDenjoy1921">Denjoy (1921)</a> 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:

<ul><li>CM([a,b]) is quasi-analytic.</li>

</mrow>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum 1/L_{j}=\infty }</annotation>

</semantics>

</math><img src="./3e67e3944b0f83244e97e5bddd27a1d2b8510e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.982ex; height:3.843ex;" alt="{\displaystyle \sum 1/L_{j}=\infty }" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}</annotation>

</semantics>

</math><img src="./08d103afa9cfd025b78850892c08634a8f36a098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.335ex; height:5.009ex;" alt="{\displaystyle L_{j}=\inf _{k\geq j}(k\cdot M_{k}^{1/k})}" loading="lazy">.</li>

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j}{\frac {1}{j}}(M_{j}^{*})^{-1/j}=\infty }</annotation>

</semantics>

</math><img src="./c8bc265c08bba218244e8b46c687462307335f89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.357ex; height:6.676ex;" alt="{\displaystyle \sum _{j}{\frac {1}{j}}(M_{j}^{*})^{-1/j}=\infty }" loading="lazy">, where Mj* is the largest log convex sequence bounded above by Mj.</li>

</mrow>

</munder>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j}{\frac {M_{j-1}^{*}}{(j+1)M_{j}^{*}}}=\infty .}</annotation>

</semantics>

</math><img src="./6345dbfaeb050aa11e198ca7fbe9c3374752d224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.97ex; height:7.509ex;" alt="{\displaystyle \sum _{j}{\frac {M_{j-1}^{*}}{(j+1)M_{j}^{*}}}=\infty .}" loading="lazy"></li></ul>

The proof that the last two conditions are equivalent to the second uses <a href="Carleman's_inequality" title="Carleman's inequality">Carleman's inequality</a>.

Example: <a href="#CITEREFDenjoy1921">Denjoy (1921)</a> pointed out that if Mn is given by one of the sequences

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

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</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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 ,}</annotation>

</semantics>

</math><img src="./1765bdd22120a10113e034581263a58c013b2d06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.177ex; height:3.009ex;" alt="{\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 ,}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading2"><h2 id="Additional_properties">Additional properties</h2></div>

For a logarithmically convex sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> the following properties of the corresponding class of functions hold:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}}</annotation>

</semantics>

</math><img src="./1204e9d455567b5994e4669052d786746f3a6236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:2.676ex;" alt="{\displaystyle C^{M}}" loading="lazy"> contains the analytic functions, and it is equal to it if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }</annotation>

</semantics>

</math><img src="./c3608066e378b2c4f13fd20fbbdc1e8d22c7bd5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.45ex; height:5.343ex;" alt="{\displaystyle \sup _{j\geq 1}(M_{j})^{1/j}<\infty }" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is another logarithmically convex sequence, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{j}\leq C^{j}N_{j}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{j}\leq C^{j}N_{j}}</annotation>

</semantics>

</math><img src="./4af9cb37257895240698bffd6c723468df8bb7a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.746ex; height:3.343ex;" alt="{\displaystyle M_{j}\leq C^{j}N_{j}}" loading="lazy"> for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{M}\subset C^{N}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}\subset C^{N}}</annotation>

</semantics>

</math><img src="./1740ab8dddb70574bdf4110c3ba4b018350ec69c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.345ex; height:2.676ex;" alt="{\displaystyle C^{M}\subset C^{N}}" loading="lazy">.</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}}</annotation>

</semantics>

</math><img src="./1204e9d455567b5994e4669052d786746f3a6236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:2.676ex;" alt="{\displaystyle C^{M}}" loading="lazy"> is stable under differentiation if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

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</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }</annotation>

</semantics>

</math><img src="./7f1bd3f71043169ef67ad047d0690a151bedd59b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.877ex; height:5.343ex;" alt="{\displaystyle \sup _{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty }" loading="lazy">.</li>

<li>For any infinitely differentiable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> there are quasi-analytic rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{M}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}}</annotation>

</semantics>

</math><img src="./1204e9d455567b5994e4669052d786746f3a6236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:2.676ex;" alt="{\displaystyle C^{M}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{N}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{N}}</annotation>

</semantics>

</math><img src="./cd8ba777375ec9e42e7bd6d8671da387e48e2936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.676ex;" alt="{\displaystyle C^{N}}" loading="lazy"> and elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in C^{M}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in C^{M}}</annotation>

</semantics>

</math><img src="./e1c357611fd2f324f492d2789053478a8af77712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.714ex; height:3.009ex;" alt="{\displaystyle g\in C^{M}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in C^{N}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in C^{N}}</annotation>

</semantics>

</math><img src="./28095b7c9ed71d95cc6c484a3b3e8e2183365e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.669ex; height:2.676ex;" alt="{\displaystyle h\in C^{N}}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g+h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=g+h}</annotation>

</semantics>

</math><img src="./3dd625168654eae3e8eaff10ccd5865ab2deb63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.672ex; height:2.509ex;" alt="{\displaystyle f=g+h}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Weierstrass_division">Weierstrass division</h3></div>

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./6032f72d4d3deedc8f4a9e49617e46b95bfc8e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.242ex; height:2.676ex;" alt="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> is said to be regular of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}}</annotation>

</semantics>

</math><img src="./df09316595abe6203a96eb1526ef693940c3ad4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.013ex; height:3.009ex;" alt="{\displaystyle g(0,x_{n})=h(x_{n})x_{n}^{d}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(0)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(0)\neq 0}</annotation>

</semantics>

</math><img src="./afb9b0ebf9fe9c7fd8ecd0c415d638af923ca322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.572ex; height:2.843ex;" alt="{\displaystyle h(0)\neq 0}" loading="lazy">. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> regular of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy">, a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation>

</semantics>

</math><img src="./730f6906700685b6d52f3958b1c2ae659d2d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" loading="lazy"> of real or complex functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> variables is said to satisfy the Weierstrass division with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in A_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in A_{n}}</annotation>

</semantics>

</math><img src="./fdd306fb26fa294a2f06294aeafd9ff13491555c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.081ex; height:2.509ex;" alt="{\displaystyle f\in A_{n}}" loading="lazy"> there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\in A}</annotation>

</semantics>

</math><img src="./ddccae96e35246ba281a43aea9d2daf5cd1d7a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.653ex; height:2.509ex;" alt="{\displaystyle q\in A}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}}</annotation>

</semantics>

</math><img src="./0fd164cb6500d85a4abedb9188068a7fb91297a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.433ex; height:2.509ex;" alt="{\displaystyle h_{1},h_{2},\ldots ,h_{d-1}\in A_{n-1}}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle f=gq+h}</annotation>

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</math><img src="./375bc60af0241fa634127455f36f31bfefb1287f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.742ex; height:2.509ex;" alt="{\displaystyle f=gq+h}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}}">

<msup>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}}</annotation>

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</math><img src="./59c2efe298087174a6168f2a6ca6450f7b878b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.206ex; height:7.676ex;" alt="{\displaystyle h(x',x_{n})=\sum _{j=0}^{d-1}h_{j}(x')x_{n}^{j}}" loading="lazy">.</dd></dl>

While the ring of analytic functions and the ring of formal <a href="Power_series" title="Power series">power series</a> both satisfy the Weierstrass division property, the same is not true for other quasi-analytic classes.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is logarithmically convex and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{M}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}}</annotation>

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</math><img src="./1204e9d455567b5994e4669052d786746f3a6236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:2.676ex;" alt="{\displaystyle C^{M}}" loading="lazy"> is not equal to the class of analytic function, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{M}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{M}}</annotation>

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</math><img src="./1204e9d455567b5994e4669052d786746f3a6236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:2.676ex;" alt="{\displaystyle C^{M}}" loading="lazy"> doesn't satisfy the Weierstrass division property with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}</annotation>

</semantics>

</math><img src="./2be4dcfbb4327f596bd47b55ddb2170ca588d287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.16ex; height:3.176ex;" alt="{\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=x_{1}+x_{2}^{2}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFCarleman1926" class="citation cs2"><a href="Torsten_Carleman" title="Torsten Carleman">Carleman, T.</a> (1926), Les fonctions quasi-analytiques, Gauthier-Villars</cite></li>

<li><cite id="CITEREFCohen1968" class="citation cs2"><a href="Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Cohen, Paul J.</a> (1968), "A simple proof of the Denjoy-Carleman theorem", <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">The American Mathematical Monthly</a>, 75 (1), Mathematical Association of America: 26–31, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2315100">10.2307/2315100</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2315100">2315100</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0225957">0225957</a></cite></li>

<li><cite id="CITEREFDenjoy1921" class="citation cs2"><a href="Arnaud_Denjoy" title="Arnaud Denjoy">Denjoy, A.</a> (1921), "Sur les fonctions quasi-analytiques de variable réelle", C. R. Acad. Sci. Paris, 173: 1329–1331</cite></li>

<li><cite id="CITEREFHörmander1990" class="citation cs2"><a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Hörmander, Lars</a> (1990), The Analysis of Linear Partial Differential Operators I, Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-00662-1</bdi></cite></li>

<li><cite id="CITEREFLeont'ev2001" class="citation cs2">Leont'ev, A.F. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quasi-analytic_class">"Quasi-analytic class"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFSolomentsev2001" class="citation cs2">Solomentsev, E.D. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Carleman_theorem">"Carleman theorem"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul></div><div><div class="zim-footer">

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