Quasi-analytic function

Let M=\{M_k\}_{k=0}^\infty 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 : \left |\frac{d^kf}{dx^k}(x) \right | \leq A^{k+1} k! M_k for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = 1 this is exactly the class of real analytic functions on [a,b]. The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and : \frac{d^k f}{dx^k}(x) = 0 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. Quasi-analytic functions of several variables For a function f:\mathbb{R}^n\to\mathbb{R} and multi-indexes j=(j_1,j_2,\ldots,j_n)\in\mathbb{N}^n, denote |j|=j_1+j_2+\ldots+j_n, and : D^j=\frac{\partial^j}{\partial x_1^{j_1}\partial x_2^{j_2}\ldots\partial x_n^{j_n}} : j!=j_1!j_2!\ldots j_n! and : x^j=x_1^{j_1}x_2^{j_2}\ldots x_n^{j_n}. Then f is called quasi-analytic on the open set U\subset\mathbb{R}^n if for every compact K\subset U there is a constant A such that : \left|D^jf(x)\right|\leq A^ for all multi-indexes j\in\mathbb{N}^n and all points x\in K. The Denjoy–Carleman class of functions of n variables with respect to the sequence M on the set U can be denoted C_n^M(U), although other notations abound. The Denjoy–Carleman class C_n^M(U) 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. Quasi-analytic classes with respect to logarithmically convex sequences In the definitions above it is possible to assume that M_1=1 and that the sequence M_k is non-decreasing. The sequence M_k is said to be logarithmically convex, if : M_{k+1}/M_k is increasing. When M_k is logarithmically convex, then (M_k)^{1/k} is increasing and : M_rM_s\leq M_{r+s} for all (r,s)\in\mathbb{N}^2. The quasi-analytic class C_n^M with respect to a logarithmically convex sequence M satisfies: • C_n^M is a ring. In particular it is closed under multiplication. • C_n^M is closed under composition. Specifically, if f=(f_1,f_2,\ldots f_p)\in (C_n^M)^p and g\in C_p^M, then g\circ f\in C_n^M. == Denjoy–Carleman theorem ==

The Denjoy–Carleman theorem, proved by after 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: • CM([a,b]) is quasi-analytic. • \sum 1/L_j = \infty where L_j= \inf_{k\ge j}(k\cdot M_k^{1/k}). • \sum_j \frac{1}{j}(M_j^*)^{-1/j} = \infty, where Mj* is the largest log convex sequence bounded above by Mj. • \sum_j\frac{M_{j-1}^*}{(j+1)M_j^*} = \infty. The proof that the last two conditions are equivalent to the second uses Carleman's inequality. Example: pointed out that if Mn is given by one of the sequences : 1,\, {(\ln n)}^n,\, {(\ln n)}^n\,{(\ln \ln n)}^n,\, {(\ln n)}^n\,{(\ln \ln n)}^n\,{(\ln \ln \ln n)}^n, \dots, then the corresponding class is quasi-analytic. The first sequence gives analytic functions. == Additional properties ==

For a logarithmically convex sequence M the following properties of the corresponding class of functions hold: • C^M contains the analytic functions, and it is equal to it if and only if \sup_{j\geq 1}(M_j)^{1/j} • If N is another logarithmically convex sequence, with M_j\leq C^j N_j for some constant C, then C^M\subset C^N. • C^M is stable under differentiation if and only if \sup_{j\geq 1}(M_{j+1}/M_j)^{1/j}. • For any infinitely differentiable function f there are quasi-analytic rings C^M and C^N and elements g\in C^M, and h\in C^N, such that f=g+h. Weierstrass division A function g:\mathbb{R}^n\to\mathbb{R} is said to be regular of order d with respect to x_n if g(0,x_n)=h(x_n)x_n^d and h(0)\neq 0. Given g regular of order d with respect to x_n, a ring A_n of real or complex functions of n variables is said to satisfy the Weierstrass division with respect to g if for every f\in A_n there is q\in A, and h_1,h_2,\ldots,h_{d-1}\in A_{n-1} such that : f=gq+h with h(x',x_n)=\sum_{j=0}^{d-1}h_{j}(x')x_n^j. While the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property, the same is not true for other quasi-analytic classes. If M is logarithmically convex and C^M is not equal to the class of analytic function, then C^M does not satisfy the Weierstrass division property with respect to g(x_1,x_2,\ldots,x_n)=x_1+x_2^2. == References ==