Search results
Results From The WOW.Com Content Network
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on L 2 and an algebra homomorphism from L 1 to L ∞, without renormalizing the Lebesgue measure. [14]
A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is Weyl's lemma. There are other weak formulations of Laplace's equation that are often useful.
Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.
A series of mathematicians applying harmonic analysis to number theory, most notably Martin Eichler, Atle Selberg, Robert Langlands, and James Arthur, have generalised the Poisson summation formula to the Fourier transform on non-commutative locally compact reductive algebraic groups with a discrete subgroup such that / has finite volume.
which is the natural normalization given by Rodrigues' formula. Plot of the spherical harmonic (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms , which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis , mathematical physics and engineering .
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians ; the solutions to which are given by eigenvalues corresponding to their modes of vibration.