Convolutions with the continuous primitive integral. (English) Zbl 1192.46039

Summary: If \(F\) is a continuous function on the real line and \(f=F'\) is its distributional derivative, then the continuous primitive integral of the distribution \(f\) is \(\int_a^bf=F(b)-F(a)\). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution \(f*g(x)=\int^\infty_{-\infty} f(x-y)g(y)\,dy\) for \(f\) an integrable distribution and \(g\) a function of bounded variation or an \(L^1\) function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For \(g\) of bounded variation, \(f*g\) is uniformly continuous and we have the estimate \(\|f*g\|_\infty\leq \|f\|\,\|g\|_{\mathcal{BV}}\), where \(\|f\|=\sup_I|\int_If|\) is the Alexiewicz norm. This supremum is taken over all intervals \(I\subset \mathbb R\). When \(g\in L^1\), the estimate is \(\|f*g\|\leq \|f\|\,\|g\|_1\). There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.


46F10 Operations with distributions and generalized functions
28B05 Vector-valued set functions, measures and integrals
26A39 Denjoy and Perron integrals, other special integrals
44A35 Convolution as an integral transform
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[1] G. B. Folland, Real Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999. · Zbl 0924.28001
[2] E. Talvila, “Henstock-Kurzweil Fourier transforms,” Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1207-1226, 2002. · Zbl 1037.42007
[3] F. G. Friedlander, Introduction to the Theory of Distributions, Cambridge University Press, Cambridge, UK, 2nd edition, 1998. · Zbl 0971.46024
[4] E. Talvila, “The distributional Denjoy integral,” Real Analysis Exchange, vol. 33, no. 1, pp. 51-82, 2008. · Zbl 1154.26011
[5] E. Talvila, “The regulated primitive integral,” Illinois Journal of Mathematics, preprint. · Zbl 1207.26018
[6] A. H. Zemanian, Distribution Theory and Transform Analysis, Dover, New York, NY, USA, 2nd edition, 1987. · Zbl 0643.46028
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York, NY, USA, 1975. · Zbl 0308.47002
[8] H. G. Dales, P. Aiena, J. Eschmeier, et al., Introduction to Banach Algebras, Operators, and Harmonic Analysis. Vol. 57, Cambridge University Press, Cambridge, UK, 2003. · Zbl 1041.46001
[9] R. M. McLeod, The Generalized Riemann Integral. Vol. 20, Mathematical Association of America, Washington, DC, USA, 1980. · Zbl 0486.26005
[10] D. D. Ang, K. Schmitt, and L. K. Vy, “A multidimensional analogue of the Denjoy-Perron-Henstock-Kurzweil integral,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 4, no. 3, pp. 355-371, 1997. · Zbl 0929.26009
[11] V. G. \vCelidze and A. G. D\vzvar\vsvili, Theory of the Denjoy Integral and Some Applications. Vol. 3, World Scientific, Singapore, 1989, translated by P. S. Bullen.
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