Orthogonalität und beste Approximation
type: CreativeWork
description: The topic of the present thesis is the determination of good approximations through orthonormal basis expansion. This method is based on fundamental mathematical ideas which transcend traditional boundaries between disciplines. In recent decades the method turned out to be exceptionally viable and (even commercially) effective in applications. To make this interesting mathematical topic more easily accessible for school teaching is the purpose of this work. At the same time I would like to present successful mathematics of the past decades and demonstrate that considering mathematical structure can be of great benefit for insight as well as applications. The subject builds on students' experience with and knowledge of elementary geometry. A fundamental perception is the fact that by dropping the perpendicular one finds the point on a given straight line or plane which is closest to a given point in the space. In combination with the ability to compute orthogonal projections by means of the inner product known from analytic geometry, this insight can be generalized and put to good use far beyond the boundaries of concrete geometry. While perpendicular lines are relevant only in two or three dimensional space, good approximations are in high demand in many areas of science and technology, even more so to handle mounds of data in the present era. High school students will be able to experience and verify in an exemplary manner how approximation methods in the Fourier analysis of audio signals or image processing in JPEG format proceed in the same way as distance calculations in geometric space. The common mathematical foundation is given by the structure of Euclidean vector spaces. The notion of 'vector space' in its most general form has taken shape during the second half of the 19th century and has proven to be extremely viable. Particularly important steps were the transition to n-dimensional space on the one hand and to function spaces on the other hand. These tran…
abstract: The topic of the present thesis is the determination of good approximations through orthonormal basis expansion. This method is based on fundamental mathematical ideas which transcend traditional boundaries between disciplines. In recent decades the method turned out to be exceptionally viable and (even commercially) effective in applications. To make this interesting mathematical topic more easily accessible for school teaching is the purpose of this work. At the same time I would like to present successful mathematics of the past decades and demonstrate that considering mathematical structure can be of great benefit for insight as well as applications. The subject builds on students' experience with and knowledge of elementary geometry. A fundamental perception is the fact that by dropping the perpendicular one finds the point on a given straight line or plane which is closest to a given point in the space. In combination with the ability to compute orthogonal projections by means of the inner product known from analytic geometry, this insight can be generalized and put to good use far beyond the boundaries of concrete geometry. While perpendicular lines are relevant only in two or three dimensional space, good approximations are in high demand in many areas of science and technology, even more so to handle mounds of data in the present era. High school students will be able to experience and verify in an exemplary manner how approximation methods in the Fourier analysis of audio signals or image processing in JPEG format proceed in the same way as distance calculations in geometric space. The common mathematical foundation is given by the structure of Euclidean vector spaces. The notion of 'vector space' in its most general form has taken shape during the second half of the 19th century and has proven to be extremely viable. Particularly important steps were the transition to n-dimensional space on the one hand and to function spaces on the other hand. These tran…
author: Heitzer, Johanna
language: ger
keywords: info:eu-repo/classification/ddc/510, Vektorraum, Euklidischer Vektorraum, Iteration, L2-Approximation, Beste-Approximation-Methode, Diskrete Approximation, Approximation, Approximationsalgorithmus, Beste Approximation, Mathematikunterricht, Didaktik, Angewandte Mathematik, Mathematik, Multiskalenanalyse, Orthogonalprojektion, Fourieranalyse, Bildverarbeitung, Oberstufe, multiresolution analysis, orthogonal projection, fourier analysis, image processing, highschool
date published: 2010
date modified: 2026-07-30T19:43:35Z
provider: RWTH Aachen University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Publikationsserver der RWTH Aachen University