×

The division problem for tempered distributions of one variable. (English) Zbl 1246.46041

The authors deal with the following division problem: suppose \(F\in C^{\infty}(\mathbb{R})\) satisfies \(F\mathcal{S}(\mathbb{R})\subset\mathcal{S}(\mathbb{R})\) (such an \(F\) is called a multiplier). For each tempered distribution \(T\in\mathcal{S}'(\mathbb{R})\) find another \(S\in\mathcal{S}'(\mathbb{R})\) so that \(T=FS\). In the first section, the authors show that a positive solution to the division problem for \(F\) is equivalent to the multiplication operator \(M_F:\mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R}),\,f\mapsto Ff\) having a closed range. In Theorem 2.1 they characterize, in terms of the properties of \(F\), when \(M_F\) has closed range. And in Theorem 2.3 they show that this is the case if and only if \(M_F\) has a linear and continuous left inverse. This last statement does no longer hold in the several variable case even for polynomials as follows from results due to M. Langenbruch [Proc. R. Soc. Edinb., Sect. A 114, No. 3–4, 169–179 (1990; Zbl 0711.35032)].

MSC:

46F10 Operations with distributions and generalized functions

Citations:

Zbl 0711.35032
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] De Vore, R. A.; Lorentz, G. G., Constructive Approximation (1993), Springer: Springer Berlin
[2] Hörmander, L., On the division of distributions, Ark. Mat., 3, 555-568 (1958) · Zbl 0131.11903
[3] Hörmander, L., The Analysis of Linear Partial Differential Operators (1983), Springer: Springer Berlin
[4] Horváth, J., Topological Vector Spaces and Distributions (1966), Addison-Wesley: Addison-Wesley Reading · Zbl 0143.15101
[5] Langenbruch, M., Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions, Proc. Roy. Soc. Edinburgh Sect. A, 114, 169-179 (1990) · Zbl 0711.35032
[6] Lojasiewicz, S., Sur le problème de la division, Studia Math., 18, 87-136 (1959) · Zbl 0115.10203
[7] Malgrange, B., Idéaux de fonctions différentiables et division des distributions, Distributions (2003), Ed. Éc. Polytech.: Ed. Éc. Polytech. Palaiseau, pp. 1-21
[8] Meise, R.; Vogt, D., Introduction to Functional Analysis (1997), Clarendon Press: Clarendon Press Oxford
[9] Schwartz, L., Théorie des Distributions (1966), Hermann: Hermann Paris
[10] Tougeron, J.-C., Idéaux de Fonctions Différentiables (1972), Springer: Springer Berlin · Zbl 0251.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.