Approximation by simple partial fractions and their generalizations
Tóm tắt
We consider a multiple interpolation by Padé simple partial fractions and propose a method for calculating the values of rational functions and polynomials on the basis of approximation by special rational functions (their numerator and denominator are represented as the differences between two simple partial fractions). We obtain an extrapolation formula for an analytic function h(z) in a neighborhood of the origin. For an extrapolation tool we use the expressions Σ
k
λ
k
h(λ
k
z), where λ
k
are calculated by a certain algorithm and are independent of the choice of h. Bibliography: 17 titles.
Tài liệu tham khảo
V. I. Danchenko and D. Ya. Danchenko, “Uniform approximation by logariphmic derivatives of polynomials” [in Russian], In: Function Theory, Applications, and Related Problems, pp. 74–77, Kazan’ (1999).
O. N. Kosukhin, “Approximation properties of the most simple fractions” [in Russian], Vestn. Mosk. Univ., Ser. I No. 4, 54–59 (2001); English transl.: Mosc. Univ. Math. Bull. 56, No.4, 36–40 (2001).
V. I. Danchenko and D. Ya. Danchenko, “Approximation by simple fractions” [in Russian], Mat. Zametki 70, No. 4, 553–559 (2001); English transl.: Math. Notes 70, No. 4, 502–507 (2001).
O. N. Kosukhin, Some Nontraditional approximation Methods Connected with Complex Polynomials, Thes. PhD, Moskov. State Univ. (2005).
V. I. Danchenko, “Approximation properties of sums of the form Σ k λ k h(λ k z)” [in Russian], Mat. Zametki 83, No. 5, 643–649 (2008); Math. Notes 83, No. 5, 587–593 (2008).
V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions” [in Russian], Mat. Sb. 197, No. 54, 33–52 (2006); Sb. Math. 197, No. 54, 505–524 (2006).
A. V. Fryantsev, “Numerical approximation of differential polynomials” [in Russian], Izv. Sarat. Univ. Ser. Mat. Mekh. Informat. 7, No. 2, 39–43 (2007).
P. V. Chunaev, “On a nontraditional method of approximation” [in Russian], Tr. Mat. Inst. Steklova 270, 281–287 (2010); English transl.: Proc. Steklov Inst. Math. 270, 278–284 (2010).
V. I. Danchenko and P. V. Chunaev, “Hermitian formulas for simple partial fractions” [in Russian], In: Modern Problems in Analysis and Teaching of Mathematics, p. 19, Moskow State Univ. Press, Moscow (2010).
V. I. Danchenko and P. V. Chunaev, “On extrapolation by λ-sums” [in Russian], In: International Conference on Differential Equations and Dynamical Systems (Suzdal’, 2010), pp. 72–73, Steklov Inst. Math., Moscow (2010).
V. I. Danchenko and P. V. Chunaev, “Approximation of polynomials by special rational fractions” [in Russian], In: Modern Methods of Function Theorem and Related Problems, pp. 115–116 Voronezh State Univ. Press, Voronezh (2011).
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E. A. Volkov, Numerical Methods, Fizmatlit, Moscow (1987).
V. L. Goncharov, Theory of INterpolation and Approximation of Functions [in Russian], GITTL, Moscow (1954),
A. P. Prudnikov, Yu. A. Bryuchkov, and O. I. Marichev, Integrals and Series. I. Elementary Functions [in Russian], Fizmatlit, Moscow (2002); English transl.: Gordon and Breach Sci. Publ., New York (1986).
A. F. Leont’ev, Entire Functions. Exponential Series [in Russian], Nauka, Moscow (1983).