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 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 are equal on an interval , then they are identical (to coincide "locally" here means equality of the functions in the interior of ). Coincidence "locally" for analytic functions can also mean equality of the functions together with all their derivatives at some point , . 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 be a sequence of positive numbers and let be some interval on the real axis. Let be the set of infinitely-differentiable functions on for which

where is a constant not dependent on . A function is analytic on if and only if for some ,

(1)

Thus, the class of analytic functions on is the class . Hadamard's problem consists in determining conditions on the numbers such that every function in the class vanishing together with each of its derivatives at some point , , is identically zero (or, what is the same, such that two functions in that are equal together with all their derivatives at a point are equal everywhere). A class with this property is called quasi-analytic on . According to what has been said above, the class is quasi-analytic on .

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

then 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 :

a) if one sets

then

b) if one sets

then

c) either

or

and

where is the convex regularization by means of logarithms of the sequence . Condition a) is called the Carleman condition, b) the Ostrowski condition, c) the Wang–Mandelbrojt condition.

For the case ( as ), one has , condition a) holds, and again one finds that is a quasi-analytic class. For the case , one has , condition a) holds, and hence the Denjoy class is quasi-analytic. In the case , , one has

as a result of which is not quasi-analytic.

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

where and do not depend on , and is the best approximation error of on by polynomials of degree . With this in mind, he considered the class of functions on satisfying the condition

(2)

where is an infinite increasing sequence of integers, and showed that if a function of this class vanishes on some interval , then it is identically zero. A class defined on is called (Bernshtein) quasi-analytic if two functions of this class that are equal on some segment are necessarily equal on the whole of . The class (2) is quasi-analytic in this sense. It should be noted that (2) does not imply that 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 and in the series

under which the class of such functions is quasi-analytic; conditions have been found on the numbers such that functions that are analytic in the disc , are infinitely differentiable on the closed disc and satisfy the conditions

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 , 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