The finite section method for infinite Vandermonde matrices and applications
Hermann Rabe · Boloka Institutional Repository (North-west University) · 2007
In this thesis we investigate a very well known and relevant question, i.e, to solve a linear equation Ax = b,where A and b are given.In our study A denotes an infinite matrix of special form called a Vandermonde matrix and b will be a vector from a given sequence space.We will consider two cases of the equation above.Different constraints will be placed upon the entries of A and b will be chosen from different sequence spaces.We will also look at an example from the first case to show how the theory can be applied.Our approach to solving this equation will be to apply the Finite Section Method.Here we follow the exposition of [9] while clarifying and explaining their approach.In addition, we will draw on various mathematical fields to assist our investigation.These include linear algebra, functional analysis, operator theory, complex analysis and topological vector spaces. OpsommingIn hierdie verhandeling ondersoek ons 'n bekende en relevante vraag, nl., die oplossing van 'n lineere vergelyking Ax = b, waar A en b gegee is.In ons studie definieer A 'n oneindige matriks van spesiale vorm, genaamd 'n Vandermonde matriks, en b is 'n vektor uit 'n gegewe ryruimte.Ons sal twee gevalle van die vergelyking hierbo beskou.Verskillende voorwaardes sal op die inskrywings van A geplaas word, terwyl b uit verskillende vektorruimtes gekies sal word.Ons sal ook 'n voorbeeld van die eerste geval beskou om te illustreer hoe die teorie toegepas word.Ons benadering tot hierdie probleem sal wees om die eindige seksiemetode toe te pas.Ons volg hier die uiteensetting van [9] terwyl ons dit volledig verduidelik.Ons sal ook gebruik maak van 'n verskeidenheid wiskundige velde om ons ondersoek te ondersteun.Dit sluit in lineere algebra, funksionaal analise, operatorteorie, komplekse analise en topologiese vektorruimtes.