Abstract
Fourier methods broadly construed have applications beyond the problems discussed in previous chapters. All of these consist, in a sense, of different decompositions for functions. The motivation for the decomposition varies from a need for efficient storage, shifted point of view, to geometrically motivated adaptations of standard transforms.
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Notes
- 1.
The author was stunned to learn that the uncertainty principle was a manifestation of the Cauchy–Schwarz inequality rather than a mystical property of subatomic particles.
- 2.
The connection of this with the quantum mechanical version comes from identifying f(t) with the wave function. Then t corresponds to the position variable, and the time variance is the “spatial uncertainty.” The quantum mechanical momentum is identified as \(\frac{h} {2\pi \,i}\, \frac{d} {dt}\,f(t)\), and our frequency variance differs by factors of Planck’s constant and 2 π from the “momentum uncertainty.”
- 3.
The statement clearly is quite imprecise as it stands, since the time function (or the Fourier transform, for that matter) contains all there is to know about the function.
- 4.
Of course, for particular windows relations can be found.
- 5.
Phonemes are a linguistic notion, corresponding to an individually identifiable speech sound segment. The spectrogram allows one to “see” the phonemes in terms of the time frequency distribution of the power in the sonic signal.
- 6.
This integral was effectively considered in Section 7.6 as a Fourier inversion example.
- 7.
The formulas occur in communication problem calculations, especially in models involving phase modulation.
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Further Reading
C.S. Burrus, R.A. Gopinath, H. Guo, Wavelets and Wavelet Transforms (Prentice-Hall, Upper Saddle River, 1998)
E.U. Condon, Quantum Mechanics (McGraw-Hill, New York, 1929)
I. Daubechies (ed.), Different Perspectives on Wavelets (American Mathematical Society, Providence, 1993)
J.H. Davis, Foundations of Deterministic and Stochastic Control (Birkhäuser, Boston, 2002)
L.I. Schiff, Quantum Mechanics, 2nd edn. (McGraw-Hill, New York, 1955)
A. Teolis, Computational Signal Processing with Wavelets (Birkhäuser, Boston, 1998)
B. Van der Pol, H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral, 2nd edn. (Cambridge University Press, Cambridge, 1955)
J.S. Walker, Wavelets and Their Scientific Applications (CRC Press LLC, Boca Raton, 1999)
D.F. Walnut, An Introduction to Wavelet Analysis (Birkhäuser, Boston, 2002)
D.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1946)
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Davis, J.H. (2016). Additional Topics. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_9
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