Debnath Lokenath - Wavelet Transforms and Their Applications

Springer Science+Business Media New York 2015

Lokenath Debnath 1 and Firdous Ahmad Shah 2

If you wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.

Henri Poincar

Historically, Joseph Fourier (17701830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonardo Euler (17071783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f ( x ) defined on the interval (,) is given by where the Fourier coefficients are In order to obtain a representation for a non-periodic function defined for all real x , it seems desirable to take limit as that leads to the formulation of the famous Fourier integral theorem: Mathematically, this is a continuous version of the completeness property of Fourier series. Physically, this form () can be resolved into an infinite number of harmonic components with continuously varying frequency and amplitude, whereas the ordinary Fourier series represents a resolution of a given function into an infinite but discrete set of harmonic components. The most significant method of solving partial differential equations in closed form, which arose from the work of P.S. Laplace (17491827), was the Fourier integral. The idea is due to Fourier, A.L. Cauchy (17891857), and S.D. Poisson (17811840). It seems impossible to assign priority for this major discovery, because all three presented papers to the Academy of Sciences of Paris simultaneously. They also replaced the Fourier series representation of a solution of partial differential equations of mathematical physics by an integral representation and thereby initiated the study of Fourier integrals. At any rate, the Fourier series and Fourier integrals, and their applications were the major topics of Fouriers famous treatise entitled Thore Analytique de la Chaleur (The Analytic Theory of Heat) published in 1822.

In spite of the success and impact of Fourier series solutions of partial differential equations, one of the major efforts, from a mathematical point of view, was to study the problem of convergence of Fourier series. In his seminal paper of 1829, P.G.L. Dirichlet (18051859) proved a fundamental theorem of pointwise convergence of Fourier series for a large class of functions. His work has served as the basis for all subsequent developments of the theory of Fourier series which was profoundly a difficult subject. G.F.B. Riemann (18261866) studied under Dirichlet in Berlin and acquired an interest in Fourier series. In 1854, he proved necessary and sufficient conditions which would give convergence of a Fourier series of a function. Once Riemann declared that Fourier was the first who understood the nature of trigonometric series in an exact and complete manner. Later on, it was recognized that the Fourier series of a continuous function may diverge on an arbitrary set of measure zero. In 1926, A.N. Kolmogorov proved that there exists a Lebesgue integrable function whose Fourier series diverges everywhere. The fundamental question of convergence of Fourier series was resolved by L. Carleson in 1966 who proved that the Fourier series of a continuous function converges almost everywhere.

In view of the abundant development and manifold applications of the Fourier series and integrals, the fundamental problem of series expansion of an arbitrary function in terms of a given set of functions has inspired a great deal of modern mathematics.

The Fourier transform originated from the Fourier integral theorem that was stated in the Fourier treatise entitled La Thore Analytique de la Chaleur , and its deep significance has subsequently been recognized by mathematicians and physicists. It is generally believed that the theory of Fourier series and Fourier transforms is one of the most remarkable discoveries in mathematical sciences and has widespread applications in mathematics, physics, and engineering. Both Fourier series and Fourier transforms are related in many important ways. Many applications, including the analysis of stationary signals and real-time signal processing, make an effective use of the Fourier transform in time and frequency domains. The Fourier transform of a signal or function f ( t ) is defined by where is a function of frequency and is the inner product in a Hilbert space. Thus, the transform of a signal decomposes it into a sine wave of different frequencies and phases, and it is often called the Fourier spectrum .

The remarkable success of the Fourier transform analysis is due to the fact that, under certain conditions, the signal f ( t ) can be reconstructed by the Fourier inverse formula Thus, the Fourier transform theory has been very useful for analyzing harmonic signals or signals for which there is no need for local information.

On the other hand, Fourier transform analysis has also been very useful in many other areas, including quantum mechanics, wave motion, and turbulence. In these areas, the Fourier transform of a function f ( x ) is defined in the space and wavenumber domains, where x represents the space variable and k is the wavenumber. One of the important features is that the trigonometric kernel