Orthogonalität und beste Approximation

  • Orthogonality and best approximation

Heitzer, Johanna; Walcher, Sebastian (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2010)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2010

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 transitions in particular play a key role in the present work. Real vector spaces in which an inner product is defined are called Euclidean. As is the case in Euclidean geometry, terms such as 'length', 'distance' and 'orthogonality' are linked to the inner product, as well as facts related to the triangle inequality or the Pythagorean theorem. Therefore, one can also transfer methods such as the determination of good approximations via orthogonal projection. In Euclidian vector spaces the orthogonal projection of a point onto a subspace provides its best approximation in this subspace, in the sense of the Euclidean norm. This projection can be decomposed into projections on mutually orthogonal, one-dimensional subspaces and thus determined via orthogonal basis expansion. These mathematical ideas stand in the focus of the present work. They can be extended to a universal method of systematic approximation or analysis of complicated mathematical objects, and can be used wherever the structure of an Euclidean vector space is given and the Euclidean norm is sensible to measure the similarity of the objects. Generally speaking, the thesis starts from a solid anchoring of concepts and relationships in elementary geometry to first lead to some applications involving column vectors in R^n, n>3, with various interpretations. For such problems the level of abstraction is moderate and alternative solution methods are available. Thus they allow to familiarize with the general method and to give an impression of its potential. Next a different view of column vectors as lists of values for piecewise constant functions is presented and emphasized. Keeping the standard inner product one has to reconsider the notion of 'orthogonality' and 'distance', as well as the question which orthogonal bases could prove useful. In the limiting case this interpretation leads to an inner product of piecewise continuous functions defined by an integral, and to the associated L^2-norm. The transition to function spaces makes applications in signal processing accessible; in this area the determination of good approximations is of particular relevance. As practically important examples the compression of digital signals via Haar wavelets and the analysis of continuous signals via Fourier expansion are treated in detail. These applications also highlight the importance of choosing subspaces in a judicious manner, and the special role of certain orthogonal systems. Within the framework of the present thesis, motivating examples, exercises and interactive Maple worksheets for all the covered topics were developed. In addition, a collection of experiments concerning the processing of optic and acoustic signals enables students to link the theoretical findings to auditory and visual perception. Theory presentation, computer materials and experiments were tested in workshops with high school students. A discussion of practicability in class and possible curriculum connections concludes the thesis.

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