119
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Sampling expansions associated with quaternion difference equations

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon
Pages 2180-2203 | Received 22 Mar 2021, Accepted 16 May 2022, Published online: 07 Jul 2022

References

  • Kramer HP. A generalized sampling theorem. J Math Phys. 1959;38(1-4):68–72.
  • Zayed AI, Hinsen G, Butzer PL. On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems. SIAM J Appl Math. 1990;50(3):893–909.
  • Zayed AI. On Kramer's sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation. SIAM J Appl Math. 1991;51(2):575–604.
  • Annaby MH. Finite Lagrange and Cauchy sampling expansions associated with regular difference equations. J Differ Equ Appl. 1998;4(6):551–569.
  • Annaby M, García A, Hernández-Medina M. On sampling and second order difference equations. Analysis. 1999;19(1):79–92.
  • García AG, Hernández-Medina MA. Discrete Sturm-Liouville problems, Jacobi matrices and Lagrange interpolation series. J Math Anal Appl. 2003;280(2):221–231.
  • Abd-Alla M, Annaby M, Hassan H, et al. Difference operators, green's matrix, sampling theory and applications in signal processing. J Differ Equ Appl. 2018;24(8):1318–1345.
  • Xu Y, Yu L, Xu H, et al. Vector sparse representation of color image using quaternion matrix analysis. IEEE Trans Image Process. 2015;24(4):1315–1329.
  • Vince J. Quaternions for computer graphics. London: Springer; 2011.
  • Campos J, Mawhin J. Periodic solutions of quaternionic-valued ordinary differential equations. Ann Mat Pura Appl. 2006;185(S5): S109–S127.
  • Wilczyński P. Quaternionic-valued ordinary differential equations. the Riccati equation. J Differ Equ. 2009;247(7):2163–2187.
  • Zhang X. Global structure of quaternion polynomial differential equations. Commun Math Phys. 2011;303(2):301–316.
  • Kou KI, Xia YH. Linear quaternion differential equations: basic theory and fundamental results. Stud Appl Math. 2018;141(1):3–45.
  • De Leo S, Ducati GC. Solving simple quaternionic differential equations. J Math Phys. 2003;44(5):2224–2233.
  • Kou KI, Liu WK, Xia YH. Solve the linear quaternion-valued differential equations having multiple eigenvalues. J Math Phys. 2019;60(2):Article ID 023510.
  • Cheng D, Kou KI, Xia YH. Floquet theory for quaternion-valued differential equations. Qual Theor Dyn Syst. 2020;19(14):1–23.
  • Li Z, Wang C, Agarwal RP, et al. Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales. Stud Appl Math. 2021;146(1):139–210.
  • Cheng D, Kou KI. Novel sampling formulas associated with quaternionic prolate spheroidal wave functions. Adv Appl Clifford Algebr. 2017;27(4):2961–2983.
  • Cheng D, Kou KI. Generalized sampling expansions associated with quaternion Fourier transform. Math Meth Appl Sci. 2018;41(11):4021–4032.
  • Wei D, Yang W, Li YM. Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain. J Frankl Inst. 2019;356(13):7571–7607.
  • Wei D, Li YM. Convolution and multichannel sampling for the offset linear canonical transform and their applications. IEEE Trans Signal Process. 2019;67(23):6009–6024.
  • Cheng D, Kou KI, Xia YH. A unified analysis of linear quaternion dynamic equations on time scales. J Appl Anal Comput. 2018;8(1):172–201.
  • Zhang F. Quaternions and matrices of quaternions. Linear Alg Appl. 1997;251:21–57.
  • Yuan SF, Wang QW. L-structured quaternion matrices and quaternion linear matrix equations. Linear Multilinear Algebra. 2016;64(2):321–339.
  • Farid FO, Nie XR, Wang QW. On the solutions of two systems of quaternion matrix equations. Linear Multilinear Algebra. 2018;66(12):2355–2388.
  • Pogorui A, Shapiro M. On the structure of the set of zeros of quaternionic polynomials. Complex Var Theory Appl Int J. 2004;49(6):379–389.
  • Janovská D, Opfer G. A note on the computation of all zeros of simple quaternionic polynomials. SIAM J Numer Anal. 2010;48(1):244–256.
  • Sudbery A. Quaternionic analysis. In: Mathematical Proceedings of the Cambridge Philosophical Society; Vol. 85. Cambridge Univ Press; 1979. p. 199–225.
  • Farid F, Wang QW, Zhang F. On the eigenvalues of quaternion matrices. Linear Multilinear Algebra. 2011;59(4):451–473.
  • Rodman L. Topics in quaternion linear algebra. Princeton (NJ): Princeton University Press; 2014. (Princeton Series in Applied Mathematics).
  • Farenick DR, Pidkowich BA. The spectral theorem in quaternions. Linear Alg Appl. 2003;371:75–102.
  • Brackx F, Delanghe R, Sommen F. Clifford analysis. Boston (MA): Pitman Advanced; 1982. (Research Notes in Mathematics; Vol. 76).
  • Opfer G. Polynomials and Vandermonde matrices over the field of quaternions. Electron Trans Numer Anal. 2009;36:9–16.
  • Cai ZF, Kou KI. Solving quaternion ordinary differential equations with two-sided coefficients. Qual Theor Dyn Syst. 2018;17(2):441–462.
  • Zou C, Kou KI, Wang Y. Quaternion collaborative and sparse representation with application to color face recognition. IEEE Trans Image Process. 2016 July;25(7):3287–3302.
  • Sangwine SJ, Le Bihan N. Quaternion toolbox for Matlabő; 2005. Software library available from: http://qtfm.sourceforge.net/
  • Annaby M, Ayad H, Rushdi M, et al. Difference operators and generalized discrete fractional transforms in signal and image processing. Signal Process. 2018;151:1–18.
  • Kang X, Tao R. Color image encryption using pixel scrambling operator and reality-preserving MPFRHT. IEEE Trans Circuits Syst Video Technol. 2019;29(7):1919–1932.
  • Li YM, Wei D, Zhang L. Double-encrypted watermarking algorithm based on cosine transform and fractional Fourier transform in invariant wavelet domain. Inf Sci. 2021;551:205–227.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.