Functional Analysis

Generalized Fourier transforms based on quantum uncertainty principle

“…For instance, function It can be shown that the spectrum of such a function will also be discrete, and the inverse integral transformation will become a sum very similar to the sum of the Fourier series. But still, with this approach, the coefficients n c of the Fourier series in (3) do not coincide with the amplitudes ) (w f of the discrete harmonics in (6). This is because of different normalizations in Fourier integral and Fourier series coefficients calculus.…”

“…Compare (6) and 7to get the following. If frequency w in (6) takes solely integer values ... , 2 , 1 , 0…”

“…. Unlike rather abstract reasoning for convergence Fourier series sum and inverse integral Fourier transform [6,7], the unitarity of the discrete transformation matrix is simply derived by elementary geometric progression and primitive complex unity root [9].…”

“…In this case, time and frequency arguments C t Î w , vary in normalized time-circle interval C . Here, DFTs (8) evolve to integral sums along unified contour C [5,6]:…”

“…The Fourier theory of heat propagation had not been recognized for a long time; later on, this approach, referred to as harmonic Fourier analysis, got various applications (Fourier series and integrals, discrete transforms, fast discrete transforms) [5]; this theory was completed after J. Fourier, when D. Hilbert became an authoritative mathematician of that time (1862-1943). The direct integral Fourier transform maps a given continuous time function whose modulus is integrated on an infinite real time axis, to its spectral image as a function of the continuous frequency, also defined on the infinite real axis, [6]. Integral transforms and Fourier series are effectively used in various fields of physics and geometry, optics, number theory, combinatory, signal processing, probability theory, etc.…”