Unified treatment of the Krätzel transformation for generalized functions. (English) Zbl 1286.46045

Summary: We discuss a generalization of the Krätzel transforms on certain spaces of ultradistributions. We prove that the Krätzel transform of an ultradifferentiable function is an ultradifferentiable function and satisfies the Parseval inequality. We also provide a complete reading of the transform constructing two desired spaces of Boehmians. Some other properties of convergence and continuity conditions and its inverse are also discussed in some detail.


46F12 Integral transforms in distribution spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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[1] Krätzel, E., Integral transformations of Bessel-type, Generalized Functions and Operational Calculus (Proc. Conf., Varna, 1975), 148-155, (1979), Sofia, Bulgaria: Bulgarian Academy of Sciences, Sofia, Bulgaria · Zbl 0403.44003
[2] Krätzel, E.; Menzer, H., Verallgemeinerte Hankel-Funktionen, Publicationes Mathematicae Debrecen, 18, 139-147, (1971) · Zbl 0247.33014
[3] Zemanian, A. H., Distribution Theory and Transform Analysis, xii+371, (1987), New York, NY, USA: Dover Publications, New York, NY, USA
[4] Cruz-Báez, D. I.; Rodríguez Expósito, J., New inversion formulas for the Krätzel transformation, International Journal of Mathematics and Mathematical Sciences, 25, 4, 253-263, (2001) · Zbl 0989.46026
[5] Zemanian, A. H., Generalized Integral Transformations, xvi+300, (1987), New York, NY, USA: Dover Publications, New York, NY, USA
[6] Barrios, J. A.; Betancor, J. J., A Krätzel’s integral transformation of distributions, Collectanea Mathematica, 42, 1, 11-32, (1991) · Zbl 0772.46019
[7] Fisher, B.; Kiliçman, A., A commutative neutrix product of ultradistributions, Integral Transforms and Special Functions, 4, 1-2, 77-82, (1996) · Zbl 0856.46023
[8] Roumieu, C., Ultra-distributions définies sur \(R^n\) et sur certaines classes de variétés différentiables, Journal d’Analyse Mathématique, 10, 153-192, (1963) · Zbl 0122.34802
[9] Roumieu, C., Sur quelques extensions de la notion de distribution, Annales Scientifiques de l’École Normale Supérieure. Troisième Série, 77, 41-121, (1960) · Zbl 0104.33403
[10] Komatsu, H., Ultradistributions. I. Structure theorems and a characterization, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 20, 25-105, (1973) · Zbl 0258.46039
[11] Beurling, A., Quasi-Analiticity and Generalized Distributions, Lectures 4 and 5, (1961), Stanford, Calif, USA: A. M. S. Summer Institute, Stanford, Calif, USA
[12] Carmichael, R. D.; Pathak, R. S.; Pilipović, S., Cauchy and Poisson integrals of ultradistributions, Complex Variables. Theory and Application, 14, 1–4, 85-108, (1990) · Zbl 0712.46018
[13] Pathak, R. S., Integral Transforms of Generalized Functions and Their Applications, xvi+415, (1997), Amsterdam, The Netherlands: Gordon and Breach Science Publishers, Amsterdam, The Netherlands
[14] Pathak, R. S., A distributional generalised Stieltjes transformation, Proceedings of the Edinburgh Mathematical Society. Series II, 20, 1, 15-22, (1976) · Zbl 0334.44004
[15] Al-Omari, S. K. Q., Certain class of kernels for Roumieu-type convolution transform of ultra-distributions of compact support, Journal of Concrete and Applicable Mathematics, 7, 4, 310-316, (2009) · Zbl 1176.46042
[16] Al-Omari, S. K. Q., Cauchy and Poison integrals of tempered ultradistributions of Roumieu and Beurling types, Journal of Concrete and Applicable Mathematics, 7, 1, 36-46, (2009) · Zbl 1190.46036
[17] Boehme, T. K., The support of Mikusiński operators, Transactions of the American Mathematical Society, 176, 319-334, (1973) · Zbl 0268.44005
[18] Al-Omari, S. K. Q.; Loonker, D.; Banerji, P. K.; Kalla, S. L., Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces, Integral Transforms and Special Functions, 19, 5-6, 453-462, (2008) · Zbl 1215.42007
[19] Mikusiński, P., Fourier transform for integrable Boehmians, The Rocky Mountain Journal of Mathematics, 17, 3, 577-582, (1987) · Zbl 0629.44005
[20] Mikusiński, P., Tempered Boehmians and ultradistributions, Proceedings of the American Mathematical Society, 123, 3, 813-817, (1995) · Zbl 0821.46053
[21] Roopkumar, R., Mellin transform for Boehmians, Bulletin of the Institute of Mathematics. Academia Sinica. New Series, 4, 1, 75-96, (2009) · Zbl 1182.46030
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