# zbMATH — the first resource for mathematics

Characterization of distributions having a value at a point in the sense of Robinson. (English) Zbl 1276.46032
In their main result, the authors give an answer to the question of how Robinson’s notion of point value is related to the classical definition of point value in the sense of S. Łojasiewicz [Stud. Math. 16, 1–36 (1957; Zbl 0086.09405)]. A distribution has a value at a point in the sense of Robinson if and only if it is a continuous function in a neighborhood of that point. Also, this result improves an earlier result of P. A. Loeb [Lect. Notes Math. 369, 153–154 (1974; Zbl 0287.26017)].

##### MSC:
 46F10 Operations with distributions and generalized functions 46S20 Nonstandard functional analysis
##### Keywords:
Schwartz distributions; nonstandard analysis; point values
Full Text:
##### References:
 [1] Robinson, A., Non-standard analysis, (1966), North-Holland Amsterdam · Zbl 0151.00803 [2] Estrada, R.; Vindas, J., A general integral, Dissertationes math., 483, 1-49, (2012) · Zbl 1245.26006 [3] Łojasiewicz, S., Sur la valeur et la limite d’une distribution en un point, Studia math., 16, 1-36, (1957) · Zbl 0086.09405 [4] Łojasiewicz, S., Sur la fixation des variables dans une distribution, Studia math., 17, 1-64, (1958) · Zbl 0086.09501 [5] Loeb, P., A note on continuity for robinson’s predistributions, (), 153-154 · Zbl 0287.26017 [6] Oberguggenberger, M., () [7] Oberguggenberger, M.; Todorov, T., An embedding of Schwartz distributions in the algebra of asymptotic functions, Int. J. math. math. sci., 21, 417-428, (1998) · Zbl 0916.46028 [8] Oberguggenberger, M.; Vernaeve, H., Internal sets and internal functions in colombeau theory, J. math. anal. appl., 341, 649-659, (2008) · Zbl 1173.46024 [9] Vernaeve, H., Nonstandard principles for generalized functions, J. math. anal. appl., 384, 536-548, (2011) · Zbl 1236.46041 [10] Estrada, R.; Kanwal, R., A distributional approach to asymptotics, (2002), Birkhäuser Boston · Zbl 0836.34056 [11] Schwartz, L., Théorie des distributions, (1966), Hermann Paris [12] Goldblatt, R., ()
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.