Harmonic analysis using methods of nonstandard analysis

Lak, Rashad Rashid Haji (2015). Harmonic analysis using methods of nonstandard analysis. University of Birmingham. Ph.D.

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Abstract

Throughout this research we use techniques of nonstandard analysis to derive and interpret results in classical harmonic analysis particularly in topological (metric) groups and theory of Fourier series.
We define monotonically definable subset \(N\) of a nonstandard *finite group \(F\), which is the monad of the neutral element of \(F\) for some invariant *metric \(d\) on \(F\). We prove some nice properties of \(N\) and the nonstandard metrisation version of first-countable Hausdorff topological groups.
We define locally embeddable in finite metric groups (LEFM). We show that every abelian group with an invariant metric is LEFM. We give a number of LEFM group examples using methods of nonstandard analysis.
We present a nonstandard version of the main results of the classical space \(L\)\(^1\)(T) of Lebesgue integrable complex-valued functions defined on the topological circle group T, to study Fourier series throughout: the inner product space; the DFT of piecewise continuous functions; some useful properties of Dirichlet and Fejér functions; convolution; and convergence in norm. Also we show the relationship between \(L\)\(^1\)(T) and the nonstandard \(L\)\(^1\)(\(F\)) via Loeb measure.
Furthermore, we model functionals defined on the test space of exponential polynomial functions on T by functionals in NSA.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):