Teoreties is ‘n onwettige piramideskema uitvoerbaar, maar wiskundigeberekeninge bewys dat dit onmoontlik is om vol te hou. Die ineenstorting van ‘n onwettige piramideskema loop gewoonlik uit op ‘n woordewisseling en hofsake. Die doel van hierdie studie is om lig te werp, eerstens oor die vraag of bedrae wat deur onwettige piramideskema verkry is ingesluit moet word by die bruto inkomste van die belastingpligtige, met spesifieke verwysing na of dit deur die belastingpligtige ontvang is. Tweedens moet gelet word dat besprekings oor onwettige piramideskemas ook die debat ontlok of belastingpligtige geregtig is op die toelaatbare aftrekkings vir uitgawes wat aangegaan is om onwettige inkomstes deur onwettige piramideskema voort te bring. Laastens sal ‘n studie gedoen word ten opsigte van die moontlikhede wat vir die belegger beskikbaar is, naamlik kan hy / sy enige aftrekkings kry vir belegging wat verlore gaan weens insolvensie van onwettige piramideskema en word hy / sy belas op inkomste verdien uit ‘n onwettige piramideskema. Die Suid-Afrikaanse regsbank worstel al vir ‘n geruime tydperk met die vraag of inkomste verdien uit onwettige aktiwiteitebelasbaar is. Een van die sterkste sosiale argumente teen die besluit om belasting te hef op inkomste verkry uit onwettige aktiwiteite, is dat die regering voordeel uit misdaad trek, deurdat hul ‘n gedeelte van bedrae wat op onwettige wyse verkry is neem. Die probleem is fundamenteel een van beleid. ‘n Breedvoerige databasis sal geskep word van regspraak en ander bronne, plaaslik sowel as internasionaal, ten einde riglyne uiteen te sit om die belasbaarheid van onwettige transaksies te bepaal, meer spesifiek die van ‘n onwettige piramideskema. / ENGLISH : An illegal pyramid scheme is theoretically feasible, but mathematical calculations have proven that it is impossible to maintain. The collapse of an illegal pyramid scheme usually ends in a heated exchange of words and lawsuits. The purpose of this study is to shed light, firstly on whether amounts obtained by an illegal pyramid scheme should be included in the gross income of the taxpayer, with specific reference to whether the taxpayer has received it. Secondly, it must be noted that the discussion of illegal pyramid schemes also produces a debate of whether the taxpayer is entitled to allowable deductions for expenses incurred to bring forward illegal revenues through an illegal pyramid scheme. Finally a study will be done regarding the available opportunities for the investor, i.e. may he / she claim for any deduction if their investment is lost due to insolvency of the illegal pyramid scheme and if he / she will be taxed on income earned out of the illegal pyramid scheme. The South African judiciary has struggled for a considerable time with the question of whether income earned from illegal activities must be taxable. One of the strongest social arguments against the decision to levy tax on income derived from illegal activities, is that the government benefits from crime, because they take a portion of the amounts that were illegally obtained. The problem is fundamentally one of policy. A comprehensive database will be created by case law and other sources, both locally and internationally, in order to set out guidelines for the taxability of illegal transactions, more specifically that of an illegal pyramid scheme. / Dissertation (LLM)--University of Pretoria, 2012. / Mercantile Law / unrestricted

在本篇論文裏，我們將使用所謂的週期化(periodization)的裝置作用於Daubechies' compactly supported wavelets上而得到一族構成L²([0,1])和H^s-periodic (the space of periodic function locally in H^s)基底的正交的週期性波包(orthonormal periodic wavelets)。然後我們給出了對於一函數的波包近似值的誤差估計（參閱定理6）以及對於週期性邊界值的常微分方程問題的解的波包近似值的誤差估計（參閱定理7）。對於Burger equation的數值解也當作一個應用來討論。 / In this thesis,we shall construct a family of orthonormal periodic wavelets which form a basis of L²([0,l]) and H^s-periodic (the space of periodic functions locally in H^s) by using a device called periodization ([10,7]) on Daubechies' compactly supported wavelets.We then give the error estimates for the wavelet approximation to a given function (see theorem 6) and to a solution of periodic boundary value problem for ordinary differential equation(see theorem 7). Numerical solution for Burger equation is also discussed as an application.