Expansion of functions of quasianalytic classes in series of polynomials
Let \(\Delta_ n(a)\) be an arbitrary domain of the n-dimensional space of the points \(x=(x_ 1,x_ 2,...,x_ n)\) that is star-shaped with respect to the point \(a=(a_ 1,a_ 2,...,a_ n)\); \(n=1,2,...<+\infty\); \(\delta_ n(a)\) be an arbitrary closed domain contained in \(\Delta_ n(a)\), \(\delta_ n(a)\ni a\); and \(\{\) M(q)\(\}\) \((q=0,1,2,...)\) be an arbitrary sequence of positive numbers. We put an infinitely differentiable function f(x), defined in \(\Delta_ n(a)\), in the class \(C_ n\{M(q)\}\), if in each domain \(\delta_ n(a)\subseteq \Delta_ n(a)\) \[ (1)\quad | D_ 1^{q_ 1}D_ 2^{q_ 2}...D_ n^{q_ n}f(x)| \leq C^{q_ 1+q_ 2+...+q_ n}M(q_ 1)M(q_ 2)....\quad M(q_ n), \] and in the class \(C_{n,1}\{M(q)\}\) if \((2)\quad | d^ qf(x)| \leq C^ qM(q).\) In equalities (1) and (2) the constant C depends on f(x) and \(\delta_ n(a)\); \(D_ i\) is the partial derivative with respect to the i-th argument; \(q_ i\) are nonnegative integers; df(x) is the differential of f(x) at the point x, and the numbers M(q) are the same for all functions of each class. The classes \(C_ n\{M(q)\}\) and \(C_{n,1}\{M(q)\}\) are said to be quasianalytic if the functions that belong to each of the classes are uniquely determined by their values and the values of all their partial derivatives at any point \(b\in \Delta_ n(a)\). Let \(\Delta_ 1(a)\) be a star-shaped (with respect to the point a) domain in the plane of the complex variable x and \(C_ 1\{q!\}\) be the class of the analytic functions f(x) of one variable x in \(\Delta_ 1(a)\). Mittag-Leffler has shown that the functions of the class \(C_ 1\{q!\}\) can be expanded in a uniformly convergent series of polynomials inside the domain \(\Delta_ 1(a):\) \[ (3)\quad f(x)=\sum^{\infty}_{\nu =0}(\sum^{p_{\nu}}_{q=0}C_{q,\nu}f^{(q)}(a)(x-a)^ q), \] where the degrees \(p_{\nu}\) of the polynomials and the numbers \(c_{q,\nu}\) are the same for all functions of the calss \(C_ 1\{q!\}\). In the present article, for the functions that belong to the quasianalytic classes \(C_ n\{M(q)\}\) and \(C_{n,1}\{M(q)\}\) we obtain expansions in uniformly and absolutely convergent series of polynomials of the same form as in Eq. (3), with the only difference that for the functions of n variables in a place of the differential \(f^{(q)}(a)(x- a)^ q\equiv d^ qf(a)\) of a function of one variable we have the differential of a function on n variables. We show that the numbers \(p_{\nu}\) and \(c_{q,\nu}\) are determined by the sequence \(\{\) M(q)\(\}\). For certain quasianalytic classes \(C_ n\{M(q)\}\), which include the classes of analytic functions of one as well as of several complex variables, we obtain expansions of form (3) for \(p_{\nu}=\nu\). This result is new even for functions of one variable. Thus, Mittag- Leffler's result is extended to more general classes of functions and is improved.
absolutely convergent series of polynomials
scientific article; zbMATH DE number 4177813