On the eigenstructure of the q-Stancu operator

Yıl 2024, Cilt: 73 Sayı: 3, 820 - 832, 27.09.2024

Övgü Gürel Yılmaz

https://doi.org/10.31801/cfsuasmas.1388792

Öz

The main goal of this research is to find the eigenvalues and the corresponding eigenfunctions of the q-Stancu operator, $L_{n,s,q}$, introduced by $L$. Yun and R. Wang. In this work, an explicit representation for moments of all orders has been derived. Further, it has been proved that $L_{n,s,q}$ possesses $n − s + 1$ linearly independent eigenfunctions whose explicit expression and the corresponding eigenvalues are derived. In addition, for special choices of parameters, several eigenfunctions are depicted.

Anahtar Kelimeler

q-integers, q-Stirling numbers, q-Stancu operators, moments, eigenvalues, eigenfunctions

Kaynakça

• Andrews, G. E., Askey, R., Roy, R., Special Functions, Encyclopedia of Mathematics and Its Applications, The University Press, Cambridge, 1999, 664 pp.
• Bernstein, S. N., Demonstration du theoreme de Weierstrass fondee sur le calcul of probabilites, Comm. Kharkov Math. Soc., 13 (1912), 1-2.
• Bostanci, T., Başcanbaz-Tunca, G., On Stancu operators depending on a non-negative integer, Filomat, 36(18) (2022), 6129-6138. https://doi.org/10.2298/FIL2218129B
• Cooper, S., Waldron, S., The eigenstructure of the Bernstein operator, J. Approx. Theory, 105(1) (2000), 133-165. https://doi.org/10.1006/jath.2000.3464
• Goodman, T. N. T., Oruç, H., Phillips, G. M., Convexity and generalized Bernstein polynomials, Proc. Edinburgh Math. Soc., 42(1) (1999), 179-190. https://doi.org/10.1017/S0013091500020101
• Gordon, W. J., Riesenfeld, R. F., Bernstein–Bezier methods for the computer-aided design of free-form curves and surfaces, J. Assoc. Comput. Mach., 21(2) (1974), 293-310. https://doi.org/10.1145/321812.321824
• Gupta, V., Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., 197(1) (2008), 172-178. https://doi.org/10.1016/j.amc.2007.07.056
• Jing, S., The q-deformed binomial distribution and its asymptotic behaviour, J. Phys. A: Math. Gen., 27(2) (1994), 493-499. https://doi.org/10.1088/0305-4470/27/2/031
• Köroğlu, B., Taşdelen Yeşildal, F., On the eigenstructure of the (α, q)-Bernstein operator, Hacet. J. Math. Stat., 50(4) (2021), 1111-1122. https://doi.org/10.15672/hujms.779544
• Landau, L. D., Lifshitz, E. M., Mechanics: Course of Theoretical Physics, Vol. 1, 3rd edition, Butterworth-Heinemann, 1976.
• Landau, L. D., Lifshitz, E. M., Quantum Mechanics: Non-Relativistic Theory 3rd Edition, Vol. 3, Butterworth-Heinemann, 1981.
• Lupaş, A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 9 (1987), 85-92.
• Ostrovska, S., Turan, M., On the eigenvectors of the q-Bernstein operators, Math. Methods Appl. Sci., 37(4) (2014), 562-570. https://doi.org/10.1002/mma.2814
• Ostrovska, S., Turan, M., On the block functions generating the limit q-Lupaş operator, Quaest. Math., 46(4) (2023), 711-719. https://doi.org/10.2989/16073606.2022.2040632
• Phillips, G. M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.
• Rajagpoal, L., Roy, S. D., Design of maximally-flat FIR filters using the Bernstein polynomial, IEEE Trans. Circuits Syst., 34(12) (1987), 1587-1590. https://doi.org/10.1109/TCS.1987.1086077
• Stancu, D. D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (1983), 211-229. https://doi.org/10.1007/BF02575593
• Xiang, X., Stancu polynomials based on the q-integers, Anal. Theory Appl., 28(3) (2012), 232-241. https://doi.org/10.3969/j.issn.1672-4070.2012.03.003
• Yun, L., Xiang, X., On shape-preserving properties and simultaneous approximation of Stancu operator, Anal. Theory Appl., 24 (2008), 195-204. https://doi.org/10.1007/s10496-008-0195-0
• Yun, L., Wang, R., Approximation and shape-preserving properties of q-Stancu operator, Anal. Theory Appl., 27 (2011), 201-210. https://doi.org/10.1007/s10496-011-0201-9
• Zee, A., Quantum Field Theory in a Nutshell, 2nd Edition, Princeton University Press, Princeton, 2003.