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 the real line or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience.
The term "harmonics" originated as the Ancient Greek word harmonikos, meaning "skilled in music". [1] In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes. Still, the term has been generalized beyond its original meaning.
Historically, Harmonic functions were the solutions of Laplace's equation, [2] this concept was extended first to special functions, [3] then to general elliptic operators [4] and nowadays harmonic functions are considered as a generalization of periodic functions [5] on function spaces defined on a manifold, for example as solutions of general, not necessarily elliptic, partial differential equations including some boundary conditions that may drive their symmetry or periodicity. [6]
The classical Fourier transform on R n 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 into the Fourier transform of f. The Paley–Wiener theorem is an example. 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 (i.e., if a signal is limited in one domain, it is unlimited in the other). This is an elementary form of an uncertainty principle in a harmonic-analysis setting.
Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation:
Abstract harmonic analysis is primarily concerned with how real or complex-valued functions (often on very general domains) can be studied using symmetries such as translations or rotations (for instance via the Fourier transform and its relatives); this field is of course related to real-variable harmonic analysis, but is perhaps closer in spirit to representation theory and functional analysis. [7]
One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is analysis on topological groups. The core motivating ideas are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups. [8]
One of the major results in the theory of functions on abelian locally compact groups is called Pontryagin duality. Harmonic analysis studies the properties of that duality. Different generalization of Fourier transforms attempts to extend those features to different settings, for instance, first to the case of general abelian topological groups and second to the case of non-abelian Lie groups. [9]
Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, the Peter–Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations. [10] This choice of harmonics enjoys some of the valuable properties of the classical Fourier transform in terms of carrying convolutions to pointwise products or otherwise showing a certain understanding of the underlying group structure. See also: Non-commutative harmonic analysis.
If the group is neither abelian nor compact, no general satisfactory theory is currently known ("satisfactory" means at least as strong as the Plancherel theorem). However, many specific cases have been analyzed, for example, SLn. In this case, representations in infinite dimensions play a crucial role.
Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components. Ocean tides and vibrating strings are common and simple examples. The theoretical approach often tries to describe the system by a differential equation or system of equations to predict the essential features, including the amplitude, frequency, and phases of the oscillatory components. The specific equations depend on the field, but theories generally try to select equations that represent significant principles that are applicable.
The experimental approach is usually to acquire data that accurately quantifies the phenomenon. For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included. In a study on vibrating strings, it is common for the experimentalist to acquire a sound waveform sampled at a rate at least twice that of the highest frequency expected and for a duration many times the period of the lowest frequency expected.
For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz. The waveform appears oscillatory, but it is more complex than a simple sine wave, indicating the presence of additional waves. The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the Fourier transform, shown in the lower figure. There is a prominent peak at 55 Hz, but other peaks at 110 Hz, 165 Hz, and at other frequencies corresponding to integer multiples of 55 Hz. In this case, 55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as harmonics.
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In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.
A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
A periodic function also called a periodic waveform, is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.
Fourier may refer to:
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a p-adic field.
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time, as in time series. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids at the frequency components of the signal. Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal.
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and spaces can be generalized.
In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects help glean properties and sometimes simplify calculations on more abstract theories.
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory.