Boris Obsieger


Approximation of Functions

Including 40 examples and 37 algorithms


Textbook at several universities

International edition in English




>> Contents (63kb pdf)

>> Algorithms, Examples, Index, References (70kb pdf)

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Numerical Methods III – Approximation of Functions


1. Introduction to Approximation

2. Least-Squares Polynomial Approximation

3. Fourier Series Expansion

4. Chebyshev Series Expansion

5. Uniform Approximation

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An excellent textbook established at several universities. Primarily written for students at technical universities, it is also a very useful handbook for engineers, PhD students and scientists. Available in several forms at all continents


uniri 50  

Appro­ved by the Univer­sity of Rijeka, Croatia, 2011-05-26.

Approved by the Faculty of Engi­neering in Rijeka, Croatia, 2011-06-27.


App­roved and recommen­­ded by the Moscow State Industrial Uni­versity, Russia, 2013-10-11.


Approved for the study programs on the international basis by the Faculty of Logi­stic in Celje, Slovenia, 2013-10-11.


This textbook introduces the reader into various types of approximations of functions, which are defined either explicitly or by their values in the distinct set of points, as well as into economisation of existing approximation formulas.


Why are the presented topics so important? Simply, because various functions cannot be calculated without approximation. Approximation formulas for some of these functions (such as trigonometric functions and logarithms) are already implemented in the calculators and standard computer libraries, providing the precision to all bits of memory in which a value is stored. So high precision is not usually required in the engineering practice, and use more numerical operations that is really necessary. Economised approximation formulas can provide required precision with less numerical operation, and can made numerical algorithms faster, especially when such formulas are used in nested loops. The other important use of approximation is in calculating functions that are defined by values in the chosen set of points, such as in solving integral equations (usually obtained from differential equations).

The book is divided into five chapters. In the first chapter, are briefly explained basic principles of approximations, i.e. approximations near the chosen point (by Maclaurin, Taylor or Padé expansion), principles of approximations with orthogonal series and principles of least squares approximations. In the second chapter, various types of least squares polynomial approximations, particularly those by using orthogonal polynomials such as Legendre, Jacobi, Laguerre, Hermite, Zernike and Gram polynomials are explained. Third chapter explains approximations with Fourier series, which are the base for developing approximations with Chebyshev polynomials (fourth chapter). Uniform approximation and further usage of Chebyshev polynomials in the almost uniform approximation, as well as in economisation of existing approximation formulas, are described in fifth chapter.


Practical application of described approximation procedures are supported by 40 examples and 37 algorithms. Besides its practical usage, the given text with 37 figures and 12 tables, partially in colour, represents a valuable background for understanding, developing and applying various numerical methods, such as interpolation, numerical integration and solving partial differential equations, which are topics in the further volumes of the series Numerical Methods.




This textbook is written in English

Several colour and black & white editions are available.



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 Reviewed by:


Prof. Maja Fosner, D.Sc.

University of Maribor, Slovenia


Prof. Damir Jelaska, D.Sc.

University of Split, Croatia


Prof. Valery Lysenko, D.Sc.

Academic of the Russian Metrological Academy

Russian Research Institute for Metrological Service, Russia


Prof. Iztok Potrc, D.Sc.

University of Maribor. Slovenia


Prof. Evgeny Pushkar, D.Sc.

Member correspondent of the Russian Academy of Natural Sciences

Moscow State Industrial University, Russia


 Previewed by:


Senior Lecturer, Katica Jurasić, M.Sc.
University of Rijeka, Croatia


 Proof reading by:


Jasenka Toplicanec, prof.

Z, Croatia


 About author:


Boris Obsieger, D.Sc., professor at the University of Rijeka, Croatia. Head of Section for Machine Elements at the Faculty of Engineering in Rijeka. Holds lectures on Machine Elements Design, Robot Elements Design, Numerical Methods in Design and Boundary Element Method. Several invited lectures. President of CADAM Conferences. Main editor of international journal Advanced Engineering. Author of several books and a lot of scientific papers.