| Title:
|
On representations of real analytic functions by monogenic functions (English) |
| Author:
|
Yuan, Hongfen |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
997-1013 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Using the method of normalized systems of functions, we study one representation of real analytic functions by monogenic functions (i.e., solutions of Dirac equations), which is an Almansi's formula of infinite order. As applications of the representation, we construct solutions of the inhomogeneous Dirac and poly-Dirac equations in Clifford \hbox {analysis}. (English) |
| Keyword:
|
monogenic function |
| Keyword:
|
inhomogeneous Dirac equation |
| Keyword:
|
inhomogeneous poly-Dirac equation |
| Keyword:
|
Almansi's formula of infinite order |
| Keyword:
|
Clifford analysis |
| MSC:
|
30G35 |
| MSC:
|
35C10 |
| MSC:
|
35J05 |
| idZBL:
|
07144870 |
| idMR:
|
MR4039615 |
| DOI:
|
10.21136/CMJ.2019.0573-17 |
| . |
| Date available:
|
2019-11-28T08:48:54Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147909 |
| . |
| Reference:
|
[1] Al-Yasiri, Z. R., Gürlebeck, K.: On a boundary value problem for a $p$-Dirac equation.Math. Methods Appl. Sci. 39 (2016), 4056-4068. Zbl 1357.30034, MR 3536522, 10.1002/mma.3847 |
| Reference:
|
[2] Aronszajn, N., Creese, T. M., Lipkin, L. J.: Polyharmonic Functions.Oxford Mathematical Monographs, Oxford University Press, Oxford (1983). Zbl 0514.31001, MR 0745128 |
| Reference:
|
[3] Bondarenko, B. A.: Operator Algorithms in Differential Equations.Izdatel'stvo Fan Uzbekskoj SSR, Tashkent (1984), Russian. Zbl 0551.34001, MR 0745129 |
| Reference:
|
[4] Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis.Research Notes in Mathematics 76, Pitman Advanced Publishing Program, Boston (1982). Zbl 0529.30001, MR 0697564 |
| Reference:
|
[5] Brackx, F., Schepper, H. De, Eelbode, D., Souček, V.: Explicit formulae for monogenic projections.Int. Conf. on Numerical Analysis and Applied Mathematics 2008 T. Simos et al. AIP Conference Proceedings 1048, American Institute of Physics, Melville (2008), 697-700. Zbl 1179.30051, 10.1063/1.2991024 |
| Reference:
|
[6] Constales, D., Grob, D., Kraußhar, R. S.: Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in $\mathbb R^{n}$ and applications to boundary value problems.J. Math. Anal. Appl. 358 (2009), 281-293. Zbl 1167.35326, MR 2532506, 10.1016/j.jmaa.2009.05.001 |
| Reference:
|
[7] Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. A Function Theory for the Dirac Operator.Mathematics and Its Applications 53, Kluwer Academic Publishers, Dordrecht (1992). Zbl 0747.53001, MR 1169463, 10.1007/978-94-011-2922-0 |
| Reference:
|
[8] Howe, R.: Remarks on classical invariant theory.Trans. Am. Math. Soc. 313 (1989), 539-570. Zbl 0674.15021, MR 0986027, 10.2307/2001418 |
| Reference:
|
[9] Kähler, U.: Clifford analysis and the Navier-Stokes equations over unbounded domains.Adv. Appl. Clifford Algebr. 11 (2001), 305-318. Zbl 1221.35280, MR 2106727, 10.1007/BF03042225 |
| Reference:
|
[10] Karachik, V. V.: Polynomial solutions to systems of partial differential equations with constant coefficients.Yokohama Math. J. 47 (2000), 121-142. Zbl 0971.35014, MR 1763777 |
| Reference:
|
[11] Karachik, V. V.: Normalized system of functions with respect to the Laplace operator and its applications.J. Math. Anal. Appl. 287 (2003), 577-592. Zbl 1039.31009, MR 2024341, 10.1016/S0022-247X(03)00583-3 |
| Reference:
|
[12] Karachik, V. V.: Method of Normalized Systems of Functions.Izd. Tsentr Yuzhno-Ural'skiĭ Gosudarstvennyĭ Universitet, Chelyabinsk (2014), Russian. Zbl 1297.35005 |
| Reference:
|
[13] Karachik, V. V.: Solution of the Dirichlet problem with polynomial data for the polyharmonic equation in a ball.Differ. Equ. 51 (2015), 1033-1042 English. Russian original translation from Differ. Uravn. 51 2015 1038-1047. Zbl 1331.35118, MR 3404090, 10.1134/S0012266115080078 |
| Reference:
|
[14] Karachik, V. V., Turmetov, B.: Solvability of some Neumann-type boundary value problems for biharmonic equations.Electron. J. Differ. Equ. 217 (2017), Paper No. 218, 17 pages. Zbl 1371.35074, MR 3711171 |
| Reference:
|
[15] Ku, M., Wang, D.: Solutions to the polynomial Dirac equations on unbounded domains in Clifford analysis.Math. Methods Appl. Sci. 34 (2011), 418-427. Zbl 1218.30138, MR 2791483, 10.1002/mma.1368 |
| Reference:
|
[16] Ryan, J.: Cauchy-Green type formulae in Clifford analysis.Trans. Am. Math. Soc. 347 (1995), 1331-1341. Zbl 0829.30030, MR 1249888, 10.2307/2154813 |
| Reference:
|
[17] Sommen, F., Jancewicz, B.: Explicit solutions of the inhomogeneous Dirac equation.J. Anal. Math. 71 (1997), 59-74. Zbl 0883.30041, MR 1454243, 10.1007/BF02788022 |
| Reference:
|
[18] Yuan, H. F.: Dirichlet type problems for Dunkl-Poisson equations.Bound. Value Probl. 2016 (2016), Article ID 222, 16 pages. Zbl 1358.30021, MR 3589586, 10.1186/s13661-016-0730-4 |
| Reference:
|
[19] Yuan, H. F.: Solutions of the Poisson equation and related equations in super spinor space.Comput. Methods Funct. Theory 16 (2016), 699-715. Zbl 1366.58008, MR 3558379, 10.1007/s40315-016-0166-y |
| Reference:
|
[20] Yuan, H. F., Karachik, V. V.: Dunkl-Poisson equation and related equations in superspace.Math. Model. Anal. 20 (2015), 768-781. MR 3427166, 10.3846/13926292.2015.1112856 |
| . |
