Taylor expansions of distributions

Abstract

In (1.18) we uglified the classical Taylor expansion of holomorphic functions in a domain Ω in the complex plane $\mathbb{C}$ , which in our context will be better denoted as ${\mathbb{R}^2}$. But it provided an understanding of what can be expected when asking for Taylor expansions of (complex-valued) functions and distributions in ${\mathbb{R}^n}$ and in domains. This resulted in the question (1.23) with (1.22) and in the considerations given afterwards in 1.4 and 1.5. We discussed this problem in Section2 for functions f belonging to spaces $B_{pq}^s\left( {{\mathbb{R}^n}} \right)$ or $F_{pq}^s\left( {{\mathbb{R}^n}} \right)$ where the smoothness s is restricted by

$$ s > {\sigma _p} = n{\left( {\frac{1}{p} - 1} \right)_ + }\user2{or }s > {\sigma _{pq}} = n{\left( {\frac{1}{{\min \left( {p,q} \right)}} - 1} \right)_ + }, $$

(8.1)

respectively. The outcome is perfect. With the β-quarks (βqu) _vm from Definition 2.4 we have

$$ f\left( x \right) = \sum\limits_{\beta,v,m} {\lambda _{vm}^\beta } {\left( {\beta qu} \right)_{vm}}\left( x \right),x \in {\mathbb{R}^n}, $$

(8.2)

where we used the notation introduced in (2.74), based on the fact that we have absolute and unconditional convergence in ${L_{\bar p}}\left( {{\mathbb{R}^n}} \right)$ with $\bar p = \max \left( {1,p} \right)$. We refer to Definition 2.6, Theorem 2.9, and the discussions in 2.7 and 2.13. In case of arbitrary smoothness s∈ℝ the situation seems to be less favourable. Instead of (8.2) we have by (3.12),

$$ f = \sum\limits_{\beta,v,m} {\left( {\eta _{vm}^\beta {{\left( {\beta qu} \right)}_{vm}} + \lambda _{vm}^\beta \left( {\beta qu} \right)_{vm}^L} \right)}, $$

(8.3)

where the β-quarks (βqu) _vm are of the same type as before, covered by Def­inition 2.4, and the β-quarks $\left( {\beta qu} \right)_{vm}^L$, introduced in Definition 3.2, satisfy some moment conditions. We refer to 3.3. By the discussion in 3.5 the series (8.3) converges unconditionally in S’(${\mathbb{R}^n}$, which justifies the formulation. On the one hand it is clear that for function spaces with, say, s < 0, moment conditions cannot be avoided. But on the other hand, one can ask whether one really needs all theη-terms in (8.3). This is not the case. The η-terms with y = 0 are sufficient and one arrives at a formulation comparable with (8.2). We discussed this situation in 3.9 with the outcome that the less ele­gant version (8.3) should be given preference in connection with applications and generalizations. However our aim in this section is different. We ask for Taylor expansions for distributions as explained above, comparable with (8.2). We concentrate on two cases, ${\mathbb{R}^n}$ and bounded C^∞ domains in ${\mathbb{R}^n}$. Function spaces serve now only as vehicles and will be simplified as much as possible. Otherwise we rely on Sections2, 3 and 5, 6 as far as ${\mathbb{R}^n}$ and bounded domains are concerned, respectively. We follow at least partly [Tri00a]. As far as ${\mathbb{R}^n}$ is concerned a few first remarks about Taylor expansions of tempered distributions may also be found in, 14.13, pp. 100–101.