Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

Investigations On The Development Of An Ann Model & Visual Manipulation Approach For 2-D Dft Computation In Image Processing

This thesis is an outcome of the investigations carried out on the development of an Artificial Neural Network (ANN) model to implement 2-D DFT at high speed. A new definition of 2-D DFT relation is presented. This new definition enables DFT computation organized in stages involving only real addition except at the final stage of computation. The number of stages is always fixed at 4. Two different strategies are proposed. 1) A visual representation of 2-D DFT coefficients. 2) A neural network approach. The visual representation scheme can be used to compute, analyze and manipulate 2D signals such as images in the frequency domain in terms of symbols derived from 2x2 DFT. This, in turn, can be represented in terms of real data. This approach can help analyze signals in the frequency domain even without computing the DFT coefficients. A hierarchical neural network model is developed to implement 2-D DFT. Presently, this model is capable of implementing 2-D DFT for a particular order N such that ((N))4 = 2. The model can be developed into one that can implement the 2-D DFT for any order N upto a set maximum limited by the hardware constraints. The reported method shows a potential in implementing the 2-D DF T in hardware as a VLSI / ASIC