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Harmonic Analysis and Representation Commons™
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- Signal processing (2)
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- Characterizing (1)
- Class Field Theory (1)
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- Fourier Transform (1)
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- Gelfand Representation (1)
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- Langland's Reciprocity (1)
- Locally Compact Abelian Groups (1)
- Multiresolution analysis (1)
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- Wavelet analysis (1)
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Articles 1 - 4 of 4
Full-Text Articles in Harmonic Analysis and Representation
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Master's Theses
Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. …
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Master's Theses
This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Master's Theses
A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …
Characterization Of The Drilling Via The Vibration Augmenter Of Rotary-Drills And Sound Signal Processing Of Impacted Pipe As A Potential Water Height Assessment Tool, Nicholas Morris
STAR Program Research Presentations
The focus of the internship has been on two topics: a) Characterize the drilling performance of a novel percussive augmenter – this drill was developed by the JPL’s Advanced Technologies Group and its performance was characterized; and b) Examine the feasibility of striking a pipe as a means of assessing the water height inside the pipe. The purpose of this investigation is to examine the possibility of using a simple method of applying impacts to a pipe wall and determining the water height from the sonic characteristic differences including damping, resonance frequencies, etc. Due to multiple variables that are relevant …