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Complex approximation by real approximation | ||
| Journal of University of Anbar for Pure Science | ||
| Volume 19, Issue 2, November and December 2025, Pages 208-215 PDF (337.82 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.37652/juaps.2024.151890.1289 | ||
| Author | ||
| Jawad Khadim Judy* | ||
| Ministry of Education- General Directorate of Babel, Iraq | ||
| Abstract | ||
| It is known that approximation of the complex functions does not, in general, depend on approximation of the real functions. However, if the complex function is viewed as a Cartesian product of two real functions, a relationship appears between the two types of approximation specified above. This means that if πΉ is a complex function where πΉ : (π₯, π¦) → (π’(π₯, π¦), (π£(π₯, π¦)), π₯, π¦ ∈ π , and π 1 , π 2 are real functions with πΉ = π 1 × π 2 , then there is a relation between an approximation of πΉ and an approximations of π 1 , π 2 . This paper studies the relation between approximation of complex functions and approximation of real functions, in terms of existence and degree, in certain special cases. The first case is when π’ is a function only of π₯ (i.e. π’(π₯)) and π£ is a function only of π¦ (i.e. π£(π¦)); this means that the complex function πΉ becomes πΉ : (π₯, π¦) → (π’(π₯), (π£(π¦)), that is, πΉ = π 1 × π 2 with π 1 : π₯ → π’(π₯) and π 2 : π¦ → π£(π¦). The second case is when there exists a relation between π₯ and π¦ in the π-π plane (i.e. √ π¦ = π (π₯)), for example π¦ = π₯ 2 or π¦ = π₯; this case leads to the first one. In addition, the relationship between the polynomial of best approximation and any polynomial of the complex function πΉ(π₯, π¦) and its analogous real functions arising from it (i.e. π 1 (π₯), π 2 (π¦)) is considered in the sense of Kolmogorov’s theorem. Furthermore, the relation between the modulus of continuity and the difference of the function πΉ(π₯, π¦) and the real functions π 1 , π 2 is examined, together with other related results. | ||
| Keywords | ||
| Approximation of complex functions; Approximation of real functions; Cartesian product of functions; Kolmogorovβs theorem | ||
| References | ||
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[1] Smirnov GS, Smirnov RG.Best Uniform Approximation of Complex-Valued Functions by Generalized Polynomials Having Restricted Ranges. Journal of Approximation Theory. 1999;100(2):284–303. 10.1006/jath.1999.3347 [2] Walsh JL.Approximation by polynomials in the complex domain.Mémorial des sciences Mathématiques, fascicule. 1935 [3] Wulbert D.Complex approximation of real functions by reciprocals of polynomials.Transactions of the American Mathematical Society. 1989;316(2):635–652.10.1090/s0002-9947-1989-0967318-5 [4] In: Lorentz GG. Approximation of Functions. Holt, Rinehart and Winston.1986; 0828403228, 9780828403221;. [5] Phillips GM, Taylor PJ.Theory and Applications of Numerical Analysis.Elsevier; 1996.10.1016/b9780-12-553560-1.x5000-9 [6] Dugundji J.Topology. Allyu and Bocan; 1966 [7] Dzyadyk VK, Shevchuk IA.Theory of Uniform Approximation of Functions by Polynomials.Walter de Gruyter; 2008.10.1515/9783110208245 | ||
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