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2018 | Buch

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Approximation Theory and Algorithms for Data Analysis

verfasst von: Prof. Armin Iske

Verlag: Springer International Publishing

Buchreihe : Texts in Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook offers an accessible introduction to the theory and numerics of approximation methods, combining classical topics of approximation with recent advances in mathematical signal processing, and adopting a constructive approach, in which the development of numerical algorithms for data analysis plays an important role.

The following topics are covered:

* least-squares approximation and regularization methods

* interpolation by algebraic and trigonometric polynomials

* basic results on best approximations

* Euclidean approximation

* Chebyshev approximation

* asymptotic concepts: error estimates and convergence rates

* signal approximation by Fourier and wavelet methods

* kernel-based multivariate approximation

* approximation methods in computerized tomography

Providing numerous supporting examples, graphical illustrations, and carefully selected exercises, this textbook is suitable for introductory courses, seminars, and distance learning programs on approximation for undergraduate students.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Contemporary applications in computational science and engineering, as well as in finance, require powerful mathematical methods to analyze big data sets. Due to the rapidly growing complexity of relevant application data at limited computational (hardware) capacities, efficient numerical algorithms are required for the simulation of complex systems with only a few parameters. Both the parameter identification and the data assimilation are based on high-performance computational methods to approximately represent mathematical functions.

Armin Iske

Chapter 2. Basic Methods and Numerical Algorithms

Abstract

In this chapter, we discuss basic mathematical methods and numerical algorithms for interpolation and approximation of functions in one variable. The concepts and principles which we address here should already be known from numerical mathematics. Nevertheless, the material of this chapter will be necessary for our subsequent discussion. Therefore, a repetition of selected elements from numerical mathematics should be most welcome.

Armin Iske

Chapter 3. Best Approximations

Abstract

In this chapter, we analyze fundamental questions of approximation.

Armin Iske

Chapter 4. Euclidean Approximation

Abstract

In this chapter, we study approximation in Euclidean spaces.

Armin Iske

Chapter 5. Chebyshev Approximation

Abstract

In this chapter, we study the problem of Chebyshev approximation.

Armin Iske

Chapter 6. Asymptotic Results

Abstract

In this chapter, we prove asymptotic statements to quantify the convergence behaviour of both algebraic and trigonometric approximation by partial sums.

Armin Iske

Chapter 7. Basic Concepts of Signal Approximation

Abstract

In this chapter, we study basic concepts of mathematical signal analysis. To this end, we first introduce the continuous Fourier transform.

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Chapter 8. Kernel-based Approximation

Abstract

This chapter is devoted to interpolation and approximation of multivariate functions.

Armin Iske

Chapter 9. Computerized Tomography

Abstract

Computerized tomography (CT) refers to a popular medical imaging method in diagnostic radiology, where large data samples are taken from a human body to generate slices of images to visualize the interior structure, e.g. of organs, muscles, brain tissue, or bones. But computerized tomography is also used in other relevant applications areas, e.g. in non-destructive evaluation of materials.

Armin Iske

Backmatter

Titel: Approximation Theory and Algorithms for Data Analysis
verfasst von: Prof. Armin Iske
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-05228-7
Print ISBN: 978-3-030-05227-0
DOI: https://doi.org/10.1007/978-3-030-05228-7