A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”

Abstract

Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.