Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 11: 26115-26133. doi: 10.3934/math.20231331
Previous Article Next Article
Research article Special Issues

Expansions of generalized bases constructed via Hasse derivative operator in Clifford analysis

  • 1.
    Mathematics Department, Faculty of Science, University of Assiut, Assiut 71516, Egypt
  • 2.
    Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
  • Received: 19 July 2023 Revised: 19 August 2023 Accepted: 27 August 2023 Published: 12 September 2023

  • MSC : 30G35, 41A10, 30D15

  • The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the $ T_{\rho} $-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Clifford settings.

    Citation: Gamal Hassan, Mohra Zayed. Expansions of generalized bases constructed via Hasse derivative operator in Clifford analysis[J]. AIMS Mathematics, 2023, 8(11): 26115-26133. doi: 10.3934/math.20231331

    Related Papers:

  • The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the $ T_{\rho} $-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Clifford settings.


    加载中


    [1] M. Abul-Ez, D. Constales, Basic sets of polynomials in Clifford analysis, Complex Var. Elliptic, 14 (1990), 177–185. https://doi.org/10.1080/17476939008814416 doi: 10.1080/17476939008814416
    [2] M. Abul-Ez, D. Constales, On the order of basic series representing Clifford valued functions, Appl. Math. Comput., 142 (2003), 575–584. https://doi.org/10.1016/S0096-3003(02)00350-8 doi: 10.1016/S0096-3003(02)00350-8
    [3] M. Abul-Ez, D. Constales, J. Morais, M. Zayad, Hadamard three-hyperballs type theorem and overconvergence of special monogenic simple series, J. Math. Anal. Appl., 412 (2014), 426–434. https://doi.org/10.1016/j.jmaa.2013.10.068 doi: 10.1016/j.jmaa.2013.10.068
    [4] B. Cannon, On the convergence of series of polynomials, Proc. London Math. Soc., 43 (1937), 348–364. https://doi.org/10.1112/plms/s2-43.5.348 doi: 10.1112/plms/s2-43.5.348
    [5] B. Cannon, On the convergence of integral functions by general basic series, Math. Z., 45 (1939), 158–205. https://doi.org/10.1007/BF01580282 doi: 10.1007/BF01580282
    [6] J. M. Whittaker, On series of poynomials, Q. J. Math., 5 (1934), 224–239. https://doi.org/10.1093/qmath/os-5.1.224 doi: 10.1093/qmath/os-5.1.224
    [7] J. M. Whittaker, Sur les séries de base polynômes quelconques, French: Gauthier-Villars, Avec la collaboration de C. Gattegno, 1949.
    [8] R. H. Makar, On derived and integral basic sets of polynomials, Proc. Amer. Math. Soc., 5 (1954), 218–225. https://doi.org/10.1090/S0002-9939-1954-0060589-3 doi: 10.1090/S0002-9939-1954-0060589-3
    [9] M. N. Mikhail, Derived and integral sets of basic sets of polynomials, Proc. Amer. Math. Soc., 4 (1953), 251–259. https://doi.org/10.1090/S0002-9939-1953-0052571-6 doi: 10.1090/S0002-9939-1953-0052571-6
    [10] M. A. Newns, On the representation of analytic functions by infinite series, Philos. T. Roy. Soc. A, 245 (1953), 429–468. https://doi.org/10.1098/rsta.1953.0003 doi: 10.1098/rsta.1953.0003
    [11] A. E. S. Ahmed, On derived and intrgated sets of basic sets of polynomials of several complex variables, Acta Math. Acad. Paed. Nyiregyhazi., 19 (2003), 195–204.
    [12] W. F. Kumuyi, M. Nassif, Derived and integrated sets of simple sets of polynomials in two complex variables, J. Approx. Theory, 47 (1986), 270–283. https://doi.org/10.1016/0021-9045(86)90018-3 doi: 10.1016/0021-9045(86)90018-3
    [13] L. Aloui, G. F. Hassan, Hypercomplex derivative bases of polynomials in Clifford analysis, Math. Meth. Appl. Sci., 33 (2010), 350–357. https://doi.org/10.1002/mma.1176 doi: 10.1002/mma.1176
    [14] M. Zayed, M. Abul-Ez, J. P. Morais, Generalized derivative and primitive of Cliffordian bases of polynomials constructed through Appell monomials, Comput. Meth. Funct. Th., 12 (2012), 501–515. https://doi.org/10.1007/BF03321840 doi: 10.1007/BF03321840
    [15] C. Hao, T. Jin, F. L. Tan, J. R. Gao, Z. X. Ma, J. Cao, The analysis of time-varying high-order moment of wind power time series, Energy Rep., 9 (2023), 3154–3159. https://doi.org/10.1016/j.egyr.2023.02.010 doi: 10.1016/j.egyr.2023.02.010
    [16] E. C. Young, Basic sets of polynomials for a generalized heat equation and its iterates, Riv. Mat. Univ. Parma., 11 (1970), 97–102.
    [17] E. P. Miles, E. Williams, Basic sets of polynomials for the iteated Laplace and wave equations, Duke Math. J., 26 (1959), 35–40. https://doi.org/10.1215/S0012-7094-59-02604-3 doi: 10.1215/S0012-7094-59-02604-3
    [18] A. Moiola, I. Perugia, A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation, Numer. Math., 138 (2018), 389–435. https://doi.org/10.1007/s00211-017-0910-x doi: 10.1007/s00211-017-0910-x
    [19] M. Abdalla, M. Abul-Ez, J. Morais, On the construction of generalized monogenic Bessel polynomials, Math. Meth. Appl. Sci., 41 (2018), 9335–9348. https://doi.org/10.1002/mma.5274 doi: 10.1002/mma.5274
    [20] M. Zayed, J. Morais, On Hadamard's three-hyperballs theorem and its applications to Whittaker-Cannon hypercomplex theory, Math. Meth. Appl. Sci., 2022, 1–14. https://doi.org/10.1002/mma.8861 doi: 10.1002/mma.8861
    [21] G. F. Hassan, M. Zayed, Equivalent base expansions in the space of Cliffodian functions, Axioms, 12 (2023), 544. https://doi.org/10.3390/axioms12060544 doi: 10.3390/axioms12060544
    [22] G. F. Hassan, M. Zayed, Approximation of monogenic functions by hypercomplex Ruscheweyh derivative bases, Complex Var. Elliptic, 2022. https://doi.org/10.1080/17476933.2022.2098279 doi: 10.1080/17476933.2022.2098279
    [23] M. Abul-Ez, D. Constales, Linear substitution for basic sets of polynomials in Clifford analysis, Port. Math., 48 (1990), 143–154.
    [24] N. Bourbaki, Topological vector classes, Springer, 1987. https://doi.org/10.1007/978-3-642-61715-7
    [25] F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Pitman research notes in math, London: Pitman, 1982.
    [26] G. F. Hassan, L. Aloui, L. Bakali, Basic sets of special monogenic polynomials in Fréchet modules, J. Complex. Anal., 2017 (2017), 2075938. https://doi.org/10.1155/2017/2075938 doi: 10.1155/2017/2075938
    [27] G. F. Hassan, L. Aloui, Bernoulli and Euler polynomials in Clifford analysis, Adv. Appl. Clifford Al., 25 (2015), 351–376. https://doi.org/10.1007/s00006-014-0511-z doi: 10.1007/s00006-014-0511-z
    [28] H. Hasse, Theorie der höheren differentiale in einem algebaischen funktionenk$\ddot{o}$rper mit vollkommenem Konstantenk $\ddot{o}$rper bei beliebiger Charakteristik, J. Reine Angew. Math., 175 (1936), 50–54. https://doi.org/10.1515/crll.1936.175.50 doi: 10.1515/crll.1936.175.50
    [29] H. Hasse, F. K. Schmidt, Noch eine Begründung der Theorie der h$\ddot{o}$heren Differentialquotienten in einem algebraischen funktionenkörper einer Unbestimmten, J. Reine Angew. Math., 177 (1937), 215–237. https://doi.org/10.1515/crll.1937.177.215 doi: 10.1515/crll.1937.177.215
    [30] O. Teichm$\ddot{u}$ller, Differentialrechung bei Charakteristik p, J. Reine Angew. Math., 175 (1936), 89–99. https://doi.org/10.1515/crll.1936.175.89 doi: 10.1515/crll.1936.175.89
    [31] M. Abul-Ez, R. D. Almeida, On the lower order and type of entire axially monogenic functions, Results Math., 63 (2013), 1257–1275. https://doi.org/10.1007/s00025-012-0267-5 doi: 10.1007/s00025-012-0267-5
    [32] Z. M. G. Kishka, A. E. S. Ahmed, On the order and type of basic and composite sets of polynomials in complete reinhardt domains, Period. Math. Hung., 46 (2003), 67–79. https://doi.org/10.1023/A:1025705824816 doi: 10.1023/A:1025705824816
    [33] M. Abul-Ez, Bessel polynomial expansions in spaces of holomorphic functions, J. Math. Anal. Appl., 221 (1998), 177–190. https://doi.org/10.1006/jmaa.1997.5840 doi: 10.1006/jmaa.1997.5840
    [34] M. Abul-Ez, M. Zayed, Criteria in Nuclear Fréchet spaces and Silva spaces with refinement of the Cannon-Whittaker theory, J. Funct. Space., 2020 (2020), 15. https://doi.org/10.1155/2020/8817877 doi: 10.1155/2020/8817877
    [35] G. F. Hassan, E. A. Salam, R. A. Rashwan, Approximation of functions by complex conformable derivative bases in Fréchet spaces, Math. Method. Appl. Sci., 46 (2022), 2636–2650. https://doi.org/10.1002/mma.8664 doi: 10.1002/mma.8664
    [36] T. Jin, F. Li, H. Peng, Uncertain barrier swaption pricing problem based on the fractional differential equation in Caputo sense, Soft Comput., 27 (2023), 11587–11602. https://doi.org/10.1007/s00500-023-08153-5 doi: 10.1007/s00500-023-08153-5
    [37] U. Kähler, N. Vieira, Fractional Clifford analysis, In: Hypercomplex analysis: New perspectives and applications, Birkhäuser, Cham, 2014. https://doi.org/10.1007/978-3-319-08771-9-13
    [38] D. Baleanu, J. E. Restrepo, D. Suragan, A class of time-fractional Dirac type operators, Chaos Soliton. Fract., 143 (2021), 110590. https://doi.org/10.1016/j.chaos.2020.110590 doi: 10.1016/j.chaos.2020.110590
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(757) PDF downloads(63) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog