The purpose of this thesis is to determine the spectrum of a type of integral operator called "Convolution Operator" acting on LP(0, °°) . In the special case p = 2 , we have a tool called the Mellin transform, this enables us to analyze the spectrum in more detail. A number of papers have dealt with special cases of our result, the reader is referred to the bibliography of a paper by D. W. Boyd; (to appear). Spectra of Convolution Operators. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis concerns the absolute convergence of the Fourier series of functions belonging to certain Lipschitz classes on totally disconnected groups. The technique used is one of interpolating between certain endpoint results which are proven directly. These results are shown to be best possible and a counterexample in interpolation theory is given. / Science, Faculty of / Mathematics, Department of / Graduate

In recent years there has been an extensive development of finite difference techniques for solution of the transient heat conduction equation due to the availability of high-speed digital computers. It is the purpose of this paper to investigate and compare several implicit methods for the solution of the Fourier equation with regards to truncation error, round-off error and computer time which probably are among the most important factors to be considered in choosing a method.

No description available.

No description available.

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1919. / Cover title. Vita. Includes bibliographical references (p. [47]).

Let {Ij} be an interval partition of the integers, f(x) a function on the circle group T and S(f) = (sum |f j|2)1/2 where fˆ j = fˆ cIj . In their 1995 paper, Hare and Klemes showed that, for fixed p ∈ (1, infinity), there exist lambdap > 1 and Ap, Bp > 0 such that if l(Ij+1)/ l(Ij) ≥ lambdap, where l(Ij) is the length of the interval Ij, then Ap∥ f∥p ≤ ∥S( f)∥p ≤ Bp∥ f∥p. That is, {Ij} is a Littlewood-Paley (p) partition. Since the intervals need not be adjacent, these partitions may be viewed as permutations of lacunary intervals. Partitions like these can be induced by subsets of sums of permuted lacunary sequences. In this thesis, we present two main results. First, complementary to the aforementioned work of Hare and Klemes who proved that sums of permuted lacunary sequences were Littlewood-Paley (p) partitions (for large enough ratio), we prove the surprising result that there are sums of permuted lacunary sequences of fixed ratio that cannot be obtained by iterating sums of permuted lacunary sequences of larger ratio finitely many times. The proof of this statement is based on the ideas developed in the 1989 paper of Hare and Klemes, especially with respect to the definition of a tree and to the theorem on the equivalency of a finitely generated partition and the absence of certain trees. These special sums may then be viewed as the critical test case for further progress on the conjecture of Hare and Klemes that sums of permuted lacunary sequences are Littlewood-Paley (p) partitions for any p. Secondly, we use the non-branching case of the method of Hare and Klemes developed in their 1992 and 1995 papers, and further developed by Hare in a general setting in 1997, to prove a result of Marcinkiewicz on iterated lacunary sequences in the case p = 4. This shows that the method introduced by Hare and Klemes can potentially be adapted to partitions other than those they were originally applied to. As well, in considering the proof given by Hare and Klemes (and by Hare in a general setting) that lacunary sequences are Littlewood-Paley (4) partitions, we present a slight variation on one of the computations which may be useful in regard to sharp versions of some of these computations, but otherwise follows the same pattern as that of the above papers. Finally, we prove an elementary property of the finite union of lacunary sequences.

This thesis is an analysis of convergence results on Fourier series. Convergence of Fourier series is studied in two ways in this thesis. The first way is in the context of Banach spaces, where the set of functions is restricted to a certain Banach space. Then the problem is in determining whether the Fourier series of a function can be represented as an element of that Banach space. The second way is in the context of pointwise convergence. Here, the problem is in determining what conditions need to be placed on an arbitrary function for its Fourier series to converge at a point.

Thesis (AE. E.D.)--Polytechnic Institute of Brooklyn, 1945. / Cover title. Reproduced from typewritten copy.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Ciência da Computação. / Made available in DSpace on 2012-10-20T10:06:07Z (GMT). No. of bitstreams: 1 192002.pdf: 536710 bytes, checksum: 9e40ddaa8a7c8e6259ff6b9eeaccb70c (MD5) / A solução de equações diferenciais nem sempre pode ser obtida em forma fechada. Em geral, faz-se necessário utilizar aproximações numéricas que tornem o problema solúvel computacionalmente. O método numérico escolhido na resolução do problema deve apresentar rápida convergência, consistência, estabilidade e baixo custo computacional. Dentre os métodos numéricos existentes para a resolução aproximada de equações diferenciais, consideramos os denominados métodos espectrais. Os métodos espectrais utilizam séries truncadas de funções suaves (infinitamente diferenciáveis) para representar a solução. Se o problema envolve dados suaves e condições de contorno periódicas, podemos conseguir uma rápida convergência (espectral) utilizando expansões em séries de Fourier. A convergência espectral é alcançada quando o erro de truncamento entre a série (com um número finito N de termos) e a solução exata, decai a zero mais rapidamente que qualquer potência de 1/N. As expansões espectrais para problemas não-periódicos (em domínios simples e finitos), geralmente utilizam séries em termos de polinômios de Chebyshev ou Legendre. Tais representações apresentam limitações quando precisamos resolver problemas transientes, pois o adensamento de pontos nodais próximo aos contornos implica na necessidade de pequenos passos no tempo para satisfazer a condição CFL.