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The objective of this paper is to construct fractional wavelet frames in L2(ℝ). A necessary condition and four sufficient conditions for fractional wavelet frames are given by virtue of fractional Fourier transform. The proposed inequalities generalize all the classical wavelet inequalities when θ = π/2. An example is presented at the end. [ABSTRACT FROM AUTHOR]

The profound study of nature is the most fertile source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means of moulding analysis itself, and discerning those elements in it which it is still essential to know and which science ought to conserve. These fundamental elements are those which recur in all natural phenomena. Joseph Fourier pure mathematics enables us to discover the concepts and laws connecting them, which gives us the key to the understanding of the phenomena of nature. Albert Einstein This article deals with a brief biographical sketch of Joseph Fourier, his first celebrated work on analytical theory of heat, his first great discovery of Fourier series and Fourier transforms. Included is a historical development of Fourier series and Fourier transforms with their properties, importance and applications. Special emphasis is made to his splendid research contributions to mathematical physics, pure and applied mathematics and his unprecedented public service accomplishments in the history of France. This is followed by historical comments about the significant and major impact of Fourier analysis on mathematical physics, probability and mathematical statistics, mathematical economics and many areas of pure and applied mathematics including geometry, harmonic analysis, signal analysis, wave propagation and wavelet analysis. Special attention is also given to the Fourier integral formula, Brownian motion and stochastic processes and many examples of applications including isoparametric inequality, everywhere continuous but nowhere differentiable functions, Heisenberg uncertainty principle, Dirichlets' theorem on primes in arithmetic progression, the Poisson summation formula and solutions of wave and diffusion equations. It is also shown that Fourier coefficients cn(t) in the Fourier expansion of a scalar field ψ(θ, t) satisfy equations of the simple harmonic motion. [ABSTRACT FROM AUTHOR]

This paper deals with a construction of wavelet frame packets on the positive half-line ℝ+ using the splitting trick for frames. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of L 2(ℝ+). If the matrix M(ξ) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3. [ABSTRACT FROM AUTHOR]

This paper deals with a construction of dyadic wavelet frames on a positive half-line + using the Walsh-Fourier transform. We prove necessary and sufficient conditions for the system [image omitted] to be a wavelet frame in L2(+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263-277], Casazza and Christenson [Weyl-Heisenberg frames for subspaces of L2(), Proc. Amer. Math. Soc. 129 (2001), pp. 145-154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273-291; An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003]. [ABSTRACT FROM AUTHOR]

Presents a historical development of wavelet transforms, Gabor transforms and their myriad of mathematical, physical, medical, biological and engineering concepts, while discussing the existence of Fourier transforms and the Hilbert space formalism in wavelet transform analysis. Factors associated with the use of the Fourier transform of a signal; Details on Gabor transforms; Historical concept of wavelet transforms.