Taylor-type expansions in terms of exponential polynomials. (English) Zbl 1528.26002
Summary: The aim of this paper is to derive an extension of the Taylor theorem related to linear differential operators with constant coefficients. For this aim, using divided differences with repeated arguments, the so-called characteristic element from the kernel of the differential operator is described. The extension of the Taylor theorem related to exponential polynomials and its consequences are established with integral remainder terms as well as in the form of mean value type theorems.
MSC:
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
| 41A05 | Interpolation in approximation theory |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
Keywords:
generalized Taylor theorem; differential operator; characteristic solution; exponential polynomialReferences:
| [1] | G. A. ANASTASSIOU,Taylor-Widder representation formulae and Ostrowski, Gr¨uss, integral means and Csiszar type inequalities, Comput. Math. Appl., 54 (1): 9-23, 2007. · Zbl 1144.26022 |
| [2] | T. M. APOSTOL,Calculus, Vol. I: One-variable calculus, with an introduction to linear algebra, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, second edition, 1967. · Zbl 0148.28201 |
| [3] | N. BOURBAKI,Functions of a Real Variable, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2004, Elementary theory, Translated from the 1976 French original [MR0580296] by Philip Spain. · Zbl 1085.26001 |
| [4] | R. L. BURDEN, J. D. FAIRES,ANDA. C. REYNOLDS,Numerical Analysis, Prindle, Weber & Schmidt, Boston, Mass., 1978. · Zbl 0419.65001 |
| [5] | M. MASJED-JAMEI, Z. MOALEMI, I. AREA,ANDJ. J. NIETO,A new type of Taylor series expansion, J. Inequal. Appl., pages Paper No. 116, 10, 2018. · Zbl 1497.41027 |
| [6] | M. MASJED-JAMEI, Z. MOALEMI, W. KOEPF,ANDH. M. SRIVASTAVA,An extension of the Taylor series expansion by using the Bell polynomials, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113 (2): 1445-1461, 2019. · Zbl 1423.41046 |
| [7] | M. OUELLET ANDS. TREMBLAY,Supersymmetric generalized power functions, J. Math. Phys., 61 (7): 072101, 19, 2020. · Zbl 1454.81098 |
| [8] | ZS. P ´ALES,A unified form of the classical mean value theorems, In Inequalities and Applications, page 493-500, World Scientific Publ. Co., River Edge, NJ, 1994. · Zbl 0882.26002 |
| [9] | ZS. P ´ALES,A general mean value theorem, Publ. Math. Debrecen, 89 (1-2): 161-172, 2016. · Zbl 1389.26013 |
| [10] | W. RUDIN,Principles of Mathematical Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-D¨usseldorf, third edition, 1976. · Zbl 0346.26002 |
| [11] | K. R. STROMBERG,Introduction to Classical Real Analysis, Wadsworth International Mathematics Series, Wadsworth International, Belmont, Calif., 1981. · Zbl 0454.26001 |
| [12] | D. V. WIDDER,A generalization of Taylor’s series, Trans. Amer. Math. Soc., 30 (1): 126-154, 1928 · JFM 54.0251.05 |
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