Quasi-analytic class

of functions

A class of functions characterized by a uniqueness property: If two functions of the class coincide "locally" , then they are identical. The simplest quasi-analytic class is the class of analytic functions on an interval $ [ a , b ] $ on the real axis (a function of this class is represented in a sufficiently small neighbourhood of each point of the interval by a Taylor series): If two analytic functions on $ [ a , b ] $ are equal on an interval $ ( \alpha , \beta ) \subset [ a , b ] $, then they are identical (to coincide "locally" here means equality of the functions in the interior of $ ( \alpha , \beta ) $). Coincidence "locally" for analytic functions can also mean equality of the functions together with all their derivatives at some point $ x _ {0} $, $ 0 \leq x _ {0} \leq b $. Coincidence "locally" in this new sense also implies equality of the functions on the whole interval.

E. Borel discovered that the uniqueness property holds not merely for analytic functions. In this connection J. Hadamard, 1912, posed the following problem. Let $ \{ M _ {n} \} $ be a sequence of positive numbers and let $ [ a , b ] $ be some interval on the real axis. Let $ C \{ M _ {n} \} $ be the set of infinitely-differentiable functions $ f $ on $ [ a , b ] $ for which

$$ | f ^ { ( n) } ( x) | \leq K ^ {n} M _ {n} ,\ \ a \leq x \leq b ,\ \ n = 0 , 1 \dots $$

where $ K = K ( f ) $ is a constant not dependent on $ n $. A function $ f $ is analytic on $ [ a , b ] $ if and only if for some $ K = K ( f ) $,

$$ \tag{1 } | f ^ { ( n) } ( x) | < K ^ {n} n ! ,\ \ a \leq x \leq b ,\ \ n = 0 , 1 ,\dots . $$

Thus, the class of analytic functions on $ [ a , b ] $ is the class $ C \{ n ! \} $. Hadamard's problem consists in determining conditions on the numbers $ M _ {n} $ such that every function in the class $ C \{ M _ {n} \} $ vanishing together with each of its derivatives at some point $ \alpha _ {0} $, $ a \leq \alpha _ {0} \leq b $, is identically zero (or, what is the same, such that two functions in $ C \{ M _ {n} \} $ that are equal together with all their derivatives at a point $ \alpha _ {0} $ are equal everywhere). A class $ C \{ M _ {n} \} $ with this property is called quasi-analytic on $ [ a , b ] $. According to what has been said above, the class $ C \{ n ! \} $ is quasi-analytic on $ [ a , b ] $.

A. Denjoy, 1921, gave sufficient conditions for quasi-analyticity. He pointed out that if

$$ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} ,\ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} ( { \mathop{\rm ln} \mathop{\rm ln} } n ) ^ {n} \dots $$

then $ C \{ M _ {n} \} $ is quasi-analytic (these classes are, by virtue of (1), wider than the class of analytic functions).

T. Carleman completely solved Hadamard's problem by giving necessary and sufficient conditions for quasi-analyticity. These conditions were subsequently modified. The Denjoy–Carleman quasi-analyticity theorem is stated as follows: Each of the following conditions is necessary and sufficient for the quasi-analyticity of the class $ C \{ M _ {n} \} $:

a) if one sets

$$ \beta _ {n} = \ \inf _ {k \geq n } M _ {k} ^ {1/k} , $$

then

$$ \sum_{n=1}^ \infty \frac{1}{\beta _ {n} } = \infty ; $$

b) if one sets

$$ T ( r) = \sup _ {n \geq 1 } \ \frac{r ^ {n} }{M _ {n} } , $$

then

$$ \int\limits _ { 0 } ^ \infty \frac{ \mathop{\rm ln} T ( r) }{r ^ {2} } \ d r = \infty ; $$

c) either

$$ \lim\limits _ {\overline{ {n \rightarrow \infty }}\; } \ M _ {n} ^ {1/n} < \infty , $$

or

$$ \lim\limits _ {n \rightarrow \infty } M _ {n} ^ {1/n} = \infty $$

and

$$ \sum ^ \infty \frac{M _ {n} ^ \prime }{M _ {n+} 1 ^ \prime } = \infty , $$

where $ \{ M _ {n} ^ \prime \} $ is the convex regularization by means of logarithms of the sequence $ \{ M _ {n} \} $. Condition a) is called the Carleman condition, b) the Ostrowski condition, c) the Wang–Mandelbrojt condition.

For the case $ M _ {n} = n ! = n ^ {n} / ( e + \epsilon _ {n} ) ^ {n} $( $ \epsilon _ {n} \rightarrow 0 $ as $ n \rightarrow \infty $), one has $ \beta _ {n} \approx n / e $, condition a) holds, and again one finds that $ C \{ n ! \} $ is a quasi-analytic class. For the case $ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} $, one has $ \beta _ {n} \approx ( n \mathop{\rm ln} n ) / e $, condition a) holds, and hence the Denjoy class $ C \{ n ( \mathop{\rm ln} n ) ^ {n} \} $ is quasi-analytic. In the case $ M = n ! ( \mathop{\rm ln} ^ {1 + \epsilon } n ) ^ {n} $, $ \epsilon > 0 $, one has

$$ \beta _ {n} \approx \ \frac{n \mathop{\rm ln} ^ {1 + \epsilon } n }{e} ,\ \ \sum_{n=1} ^ \infty \frac{1}{\beta _ {n} } < \infty , $$

as a result of which $ C \{ M _ {n} \} $ is not quasi-analytic.

S.N. Bernshtein introduced other quasi-analytic classes of functions. He showed that a function $ f $ is analytic on an interval $ [ a , b ] $ if and only if

$$ E _ {n} ( f ) < M \rho ^ {n} ,\ \ n = 0 , 1 \dots \ \rho < 1 , $$

where $ M = M ( f ) $ and $ \rho = \rho ( f ) $ do not depend on $ n $, and $ E _ {n} ( f ) $ is the best approximation error of $ f $ on $ [ a , b ] $ by polynomials of degree $ n $. With this in mind, he considered the class of functions $ f $ on $ [ a , b ] $ satisfying the condition

$$ \tag{2 } E _ {n} ( f ) < M \rho ^ {n} ,\ \rho < 1 ,\ n= n _ {1} , n _ {2} \dots $$

where $ n _ {1} , n _ {2} ,\dots $ is an infinite increasing sequence of integers, and showed that if a function of this class vanishes on some interval $ ( \alpha , \beta ) \subset [ a , b ] $, then it is identically zero. A class $ C $ defined on $ [ a , b ] $ is called (Bernshtein) quasi-analytic if two functions of this class that are equal on some segment $ ( \alpha , \beta ) \subset [ a , b ] $ are necessarily equal on the whole of $ [ a , b ] $. The class (2) is quasi-analytic in this sense. It should be noted that (2) does not imply that $ f $ is infinitely differentiable (there are appropriate examples).

Other problems of quasi-analyticity have also been studied. For example, the question has been solved concerning the rate of decrease of the coefficients $ a _ {n} $ and $ b _ {n} $ in the series

$$ f(x) = \sum_{n=0}^\infty ( a_n \cos n x + b_n \sin n x ) $$

under which the class of such functions is quasi-analytic; conditions have been found on the numbers $ M _ {n} $ such that functions $ f ( z) $ that are analytic in the disc $ | z | < 1 $, are infinitely differentiable on the closed disc $ | z | \leq 1 $ and satisfy the conditions

$$ | f ^ { ( n) } ( x) | \leq K ^ {n} M _ {n} ,\ \ n = 0 , 1 \dots \ | z | \leq 1 , $$

form a quasi-analytic class; etc.

References

[1]	S.N. Bernshtein, "Collected works" , 2 , Moscow (1964) (In Russian)
[2]	S. Mandelbrojt, "Séries de Fourier et classes quasi-analytiques de fonctions" , Gauthier-Villars (1935)
[3]	S. Mandelbrojt, "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars (1952)

Comments

The original paper by Denjoy is [a4], for Carleman's work see also [a5]. For a neat proof of the Denjoy–Carleman theorem see [a1].

Quasi-analytic classes have also been introduced on certain arcs in $ \mathbf C $, and the sufficiency part of the Denjoy–Carleman survives in this setting, cf. [a3].

References

[a1]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1987) pp. 24
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. Chapt. 1
[a3]	R. Zeinstra, "Müntz–Szász approximation on curves and area problems for zero sets" , Univ. Amsterdam (1985) (Thesis)
[a4]	A. Denjoy, "Sur les fonctions quasi-analytiques de variable réelle" C.R. Acad. Sci. Paris , 173 (1921) pp. 1329–1331
[a5]	T. Carleman, "Les fonctions quasi-analytiques" , Gauthier-Villars (1926)
[a6]	S. Mandelbrojt, "Analytic functions and classes of infinitely differentiable functions" , Pamphlet , 29 : 1 , Rice Univ. (1942)
[a7]	A. Beurling, "Quasi-analyticity" , Collected works , I , Birkhäuser (1989) pp. 396–431