Fisher, Brian; Ege, İnci; Özçaḡ, Emin On the neutrix composition of the distributions \(x^{-s} \ln^m |x|\) and \(x^r\). (English) Zbl 1194.46061 Appl. Anal. 89, No. 3, 365-375 (2010). Summary: Let \(F\) be a distribution and let \(f\) be a locally summable function. The distribution \(F(f)\) is defined as the neutrix limit of the sequence \(\{F_n (f)\}\), where \(F_n (x) = F(x) * \delta _n (x)\) and \(\{ \delta _n (x)\}\) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function \(\delta (x)\). The composition of the distributions \(x^{-s} \ln^m |x|\) and \(x^r\) is proved to exist and be equal to \(r^m x^{-rs} \ln^m |x|\) for \(r,s,m = 2, 3\dots \). Cited in 1 Document MSC: 46F10 Operations with distributions and generalized functions Keywords:distribution; delta-function; composition of distributions; neutrix; neutrix limit PDFBibTeX XMLCite \textit{B. Fisher} et al., Appl. Anal. 89, No. 3, 365--375 (2010; Zbl 1194.46061) Full Text: DOI References: [1] DOI: 10.1006/jsvi.1999.2580 · Zbl 1235.76147 [2] Antosik P, The Sequential Approach (1973) [3] Fisher B, Publ. Math. Debrecen 35 pp 37– (1988) [4] DOI: 10.1002/(SICI)1099-1476(19960910)19:13<1017::AID-MMA723>3.0.CO;2-2 · Zbl 0854.41029 [5] van der Corput JG, J. Anal. Math. 7 pp 291– (1959) [6] DOI: 10.1002/mana.19821060123 · Zbl 0499.46024 [7] DOI: 10.1002/mana.19821080110 · Zbl 0522.46025 [8] Kau H, Publ. Math. Debrecen 40 pp 279– (1992) [9] DOI: 10.1002/mana.19921570120 · Zbl 0781.46028 [10] DOI: 10.1016/S0893-9659(00)00171-3 · Zbl 0990.46030 [11] DOI: 10.1016/j.jmaa.2006.02.068 · Zbl 1112.46033 [12] Gel’fand IM, Generalized Functions (1964) [13] DOI: 10.1007/BF02843535 · Zbl 0936.46030 [14] Fisher B, Thai J. Math. 1 pp 39– (2003) [15] Fisher B, Hacettepe J. Math. Stat. 37 pp 1– (2008) 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.