Quasipower Series and Quasianalytic Classes of Functions
American Mathematical Soc., Nov 12, 2002
A certain class of functions $C$ on an interval is called quasianalytic if any function in $C$ is uniquely determined by the values of its derivatives at any point. The obvious question, then, is how to reconstruct such a function from the sequence of values of its derivatives at a certain point. In order to answer that question, Badalyan combines a study of expanding functions in generalized factorial series with a study of quasipower series. The theory of quasipower series and its application to the reconstruction problem are explained in detail in this research monograph. Along the way other, related problems are solved, such as Borel's hypothesis that no quasianalytic function can have all positive derivatives at a point. While the treatment is technical, the theory is developed chapter by chapter in detail, and the first chapter is of an introductory nature. The quasipower series technique explained here provides the means to extend the previously known results and elucidates their nature in the most relevant manner. This method also allows for thorough investigation of numerous problems of the theory of functions of quasianalytic classes by graduate students and research mathematicians.
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Abel criterion Abel transformation abscissa of convergence absolute convergence Acad admit an expansion analytic function apply the Abel arbitrary number asymptotic series Badalyan basic interval belongs Borel bounded Chapter class of functions coefficients completes the proof conditions of Theorem contour converges absolutely converges uniformly defined Definition Denote differentiation divergent domain encloses neighborhoods equality estimate exists factorial series following inequality holds function f(z functions of quasianalytic given sequence half-plane hence implies infinitely-differentiable functions integral 1.2 integrand ip(t ip(x Lemma limsup Mandelbrojt 1952 monotone functions monotone increasing necessary and sufficient obtain Obviously order of uniform proof of Theorem quasianalytic class quasianalytic function quasipower series representation right-hand side satisfies condition satisfies the conditions Section sense of 2.2 sequence 4.2 sequence mn stationary point sufficient condition Taylor formula term-by-term Theorem 2.1 tp(t tp(x uniform convergence unique Vallee-Poussin Watson's problem