The continuous partial-wave decomposition of a function over the full real line constitutes the Fourier analysis of the function. The precise formulation of this decomposition, a broad outline of its range of applicability, and its vector space aspects constitute Section 7.1. Its main properties are given in Section 7.2. Section 7.3 proceeds toward applications by the introduction of the Dirac S and its role in finding the Green’s function, which determines the time development of diffusive and elastic systems with source or driving-force terms. Except for a few connections, the following three sections are independent of each other. Section 7.4 deals with functions which have support (i.e., are not necessarily zero) on half-infinite or finite intervals. The former are interesting in that they can be used to describe causal processes. The Fourier transforms of these functions satisfy certain dispersion relations due to their behavior in the complex plane. Subtractions for band-absorption filters are described. Section 7.5 deals with the quantum oscillator wave functions. The harmonic oscillator wave functions constitute a denumerable complete and orthonormal basis for the space of square-integrable functions. The repulsive oscillator functions, on the other hand, though less well known, serve both as a generalized basis for that space and as a fine working ground for various Fourier analysis techniques. Finally, Section 7.6 describes a type of complementarity between a function and its Fourier transform which gives rise to the Heisenberg uncertainty relation between the dispersion in measurement of two quantum-mechanical observables.
KeywordsFourier Transform Harmonic Oscillator Coherent State Inverse Fourier Transform Relate Integral
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