Properties of the convolution operation in the complexity space and its dual

Authors

  • José M. Hernández-Morales Departamento de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Huajuapan de León, Oaxaca, C.P. 69000, Mexico
  • Netzahualcóyotl C. Castañeda-Roldán Universidad Tecnológica de la Mixteca, Huajuapan de León, Oaxaca, C.P. 69000, Mexico
  • Luz C. Álvarez-Marín Departamento de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Huajuapan de León, Oaxaca, C.P. 69000, Mexico

DOI:

https://doi.org/10.33044/revuma.3402

Abstract

We give the basic properties of discrete convolution in the space of complexity functions and its dual space. Two inequalities are identified, and defined in the general context of an arbitrary binary operation in any weighted quasi-metric space. In that setting, some quasi-metric and convergence consequences of those inequalities are proven. Using convolution, we show a method for building improver functionals in the complexity space. We also consider convolution in three topologies within the dual space, obtaining two topological monoids.

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References

H. Aimar, B. Iaffei, and L. Nitti, On the Macías-Segovia metrization of quasi-metric spaces, Rev. Un. Mat. Argentina 41 no. 2 (1998), 67–75.  MR  Zbl

T. Banakh and A. Ravsky, Quasi-pseudometrics on quasi-uniform spaces and quasi-metrization of topological monoids, Topology Appl. 200 (2016), 19–43.  DOI  MR  Zbl

Ş. Cobzaş, Functional analysis in asymmetric normed spaces, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2013.  DOI  MR  Zbl

A. Császár, Foundations of general topology, A Pergamon Press Book, The Macmillan Company, New York, 1963.  MR  Zbl

P. Fletcher and W. Hunsaker, Symmetry conditions in terms of open sets, Topology Appl. 45 no. 1 (1992), 39–47.  DOI  MR  Zbl

P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics 77, Marcel Dekker, New York, 1982.  DOI  MR  Zbl

L. M. García-Raffi, S. Romaguera, and M. P. Schellekens, Applications of the complexity space to the general probabilistic divide and conquer algorithms, J. Math. Anal. Appl. 348 no. 1 (2008), 346–355.  DOI  MR  Zbl

D. N. Georgiou, A. C. Megaritis, and S. P. Moshokoa, Small inductive dimension and Alexandroff topological spaces, Topology Appl. 168 (2014), 103–119.  DOI  MR  Zbl

J. Goubault-Larrecq, Non-Hausdorff topology and domain theory, New Mathematical Monographs 22, Cambridge University Press, Cambridge, 2013.  DOI  MR  Zbl

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics, second ed., Addison-Wesley, Reading, MA, 1994.  MR  Zbl

B. Iaffei and L. Nitti, A unified point of view on boundedness of Riesz type potentials, Rev. Un. Mat. Argentina 59 no. 1 (2018), 99–121.  DOI  MR  Zbl

M. İlkhan and E. E. Kara, On statistical convergence in quasi-metric spaces, Demonstr. Math. 52 no. 1 (2019), 225–236.  DOI  MR  Zbl

H.-P. A. Künzi, Nonsymmetric topology, in Topology with applications (Szekszárd, 1993), Bolyai Soc. Math. Stud. 4, János Bolyai Math. Soc., Budapest, 1995, pp. 303–338.  MR  Zbl

S. G. Matthews, Partial metric topology, in Papers on general topology and applications (Flushing, NY, 1992), Ann. New York Acad. Sci. 728, New York Acad. Sci., New York, 1994, pp. 183–197.  DOI  MR  Zbl

L. Nachbin, Topology and order, Van Nostrand Mathematical Studies, No. 4, D. Van Nostrand, Princeton, N.J., 1965.  MR  Zbl

T. Richmond, General topology: an introduction, De Gruyter Textbook, De Gruyter, Berlin, 2020.  DOI  MR  Zbl

S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory 103 no. 2 (2000), 292–301.  DOI  MR  Zbl

S. Romaguera and M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 no. 1-3 (1999), 311–322.  DOI  MR  Zbl

W. Rudin, Principles of mathematical analysis, third ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1976.  MR  Zbl

M. Schellekens, The Smyth completion: a common foundation for denotational semantics and complexity analysis, in Mathematical foundations of programming semantics (New Orleans, LA, 1995), Electron. Notes Theor. Comput. Sci. 1, Elsevier, Amsterdam, 1995, pp. 535–556.  DOI  MR  Zbl

M. Schellekens, The Smyth completion: a common topological foundation for denotational semantics and complexity analysis, Ph.D. thesis, Carnegie Mellon University, 1995.

D. Scott, Outline of a mathematical theory of computation, Tech. Monograph PRG-2, Oxford University Computing Laboratory, Programming Research Group, 1970. Available at https://www.cs.ox.ac.uk/publications/publication3720-abstract.html.

D. Scott and C. Strachey, Toward a mathematical semantics for computer languages, Tech. Monograph PRG-6, Oxford University Computing Laboratory, Programming Research Group, 1971. Available at https://www.cs.ox.ac.uk/publications/publication3723-abstract.html.

M. B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc. (2) 49 no. 2 (1994), 385–400.  DOI  MR  Zbl

P. Sünderhauf, The Smyth-completion of a quasi-uniform space, in Semantics of programming languages and model theory (Schloß Dägstuhl, 1991), Algebra Logic Appl. 5, Gordon and Breach, Montreux, 1993, pp. 189–212.  MR  Zbl

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Published

2024-05-21

Issue

Vol. 67 No. 1 (2024)

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Article

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Copyright (c) 2024 José M. Hernández-Morales, Netzahualcoyotl, Luz

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