Matris Formda Kompleks Değerli Kısmi Diferensiyel Denklemlerin Çözümleri Yardımıyla Kompleks Matris Değerli Fonksiyonlara Yaklaşım Üzerine

Year 2018, Volume: 22 Issue: 3, 1169 - 1174, 20.09.2018

Yeşim Sağlam Özkan Sezayi Hızlıyel

https://doi.org/10.19113/sdufenbed.470192

Abstract

Bu çalışmada

$A =\alpha^{(0)}+\alpha ^{(1)}\phi +\alpha^{(2)}\overline{\phi}$,
$B =\beta ^{(0)}+\beta ^{(1)}\phi +\beta ^{(2)}\overline{\phi}$,

lineer katsayılara sahip olan

$ \frac{\partial w}{\partial \bar{\phi}}=Aw+B\overline{w}$   

denkleminin çözümleri araştırıldı. Bu çözümler kullanılarak $w=K^{(0)}+\phi K^{(1)} +\bar{\phi} K^{(2)}$ formuna sahip kompleks matris değerli fonksiyonlara yaklaşıldı. Burada $\phi$, $Q$-holomorf fonksiyonlar için bir doğurucu çözümdür.

Keywords

Genelleştirilmiş Beltrami sistemleri, Genelleştirilmiş Q-holomorf fonksiyonlar, Weierstrass-Stone yaklaşım teoremi

On the Approximation to Complex Matrix-valued Functions by Using Solutions of Partial Complex Differential Equation in Matrix Form

Abstract

In this work, we seek  the solutions of the equation  

$\frac{\partial w}{\partial \bar{\phi}}=Aw+B\overline{w}$

with linear coefficients

$A=\alpha^{(0)}+\alpha ^{(1)}\phi +\alpha^{(2)}\overline{\phi}$,
$B=\beta ^{(0)}+\beta ^{(1)}\phi +\beta ^{(2)}\overline{\phi}$,

such that using this solutions we approximated to complex matrix valued function which possess the form $w=K^{(0)}+\phi K^{(1)} +\bar{\phi} K^{(2)}$. Here $\phi$ is a generating solution for $Q$-holomorphic functions.

Keywords

Generalized Beltrami systems, Generalized Q-holomorphic functions, Weierstrass-Stone approximation

References

• [1] Douglis, A. 1953. A Function theoretic approach to elliptic systems of equations in two variables. Comm. Pure Appl. Math. 6(1953), 259-289.
• [2] Bojarskiı, B. V. 1966. Theory of generalized analytic vectors (in Russian), Ann. Polon. Math. 17(1966), 281-320.
• [3] Hile, G. N. 1982. Function Theory for Generalized Beltrami Systems. Comp. Math. 11(1982), 101-125.
• [4] Hızlıyel, S., Çağlıyan, M. 2010. The Riemann Hilbert problem for generalized Q-holomorphix functions. Turk J. Math. 34(2010), 167-180.
• [5] Hızlıyel, S., Çağlıyan M. 2004. Generalized Qholomorphic functions. Complex Var. Theory Appl. 49(2004), 427-447.
• [6] Hızlıyel, S., Çağlıyan, M. 2004. Pseudo Qholomorphic functions. Complex Var. Theory Appl. 49(2004), 941-955.
• [7] Vekua, I. N. 1962. Generalized Analytic Functions.Pergamon; Oxford. 668p.
• [8] Bers, L. 1953. Theory of Pseudo-analytic Functions. New York.
• [9] Tutschke, W., Vasudeva, H. 2002. Linear generalized analytic functions. Complex Var. Theory Appl. 47(2002), 719-725.
• [10] Tutschke,W. 1989. Solution of initial value problems in classes of generalized analytic functions. Teunber Leibzig and Springer-Verlag. 188p.
• [11] Hızlıyel, S., Sağlam Özkan, Y. 2016. The Cauchy-Kowalewski theorem in the space of pseudo Qholomorphic functions, Complex Anal. Oper. Theory. 10(2016), 953-963.