Harmonic analysis

Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics", hence the name "harmonic analysis." In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience.

Related Topics:
Mathematics - Fourier series - Fourier transform - Harmonic - Signal processing - Quantum mechanics - Neuroscience

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The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley-Wiener theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very elementary form of an uncertainty principle in a harmonic analysis setting. See also classic harmonic analysis.

Related Topics:
Tempered distribution - Paley-Wiener theorem - Distribution - Compact support - Uncertainty principle - Classic harmonic analysis

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Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.

Related Topics:
Hilbert space - Functional analysis

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