Harmonic analysis on Euclidean spaces

Start here

• ConceptHarmonic analysis

  Harmonic analysis studies functions by writing them as superpositions of basic waves, and studies operators through their action on these waves. Its classical core is the theory of Fourier series on the circle and of the Fourier transform on $\R^n$. On a general locally compact group the waves are replaced by characters or irreducible representations, which gives harmonic analysis on groups. The subject supplies tools for partial differential equations, number theory, probability and signal processing.

• ConceptFourier analysis

  Fourier analysis is the study of how functions are represented by sums or integrals of sines, cosines and complex exponentials. Its main tools are the Fourier series of periodic functions, the Fourier transform on $\R^n$ and the discrete Fourier transform on $\Z / N\Z$. It is the classical core of harmonic analysis. It turns differentiation into multiplication and convolution into pointwise product, which makes it central for differential equations and signal processing.

• ConceptTrigonometric polynomial

  A trigonometric polynomial is a finite linear combination of the functions $e^{inx}$, $n \in \Z$, or equivalently of $1$, $\cos nx$ and $\sin nx$. Trigonometric polynomials are the partial sums of Fourier series. They are dense in the continuous $2\pi$-periodic functions, so they play for periodic functions the role that ordinary polynomials play on a closed interval.

• ConceptTrigonometric series

  A trigonometric series is an infinite series $\sum_{n \in \Z} c_n e^{inx}$, or $\frac{a_0}{2} + \sum_{n \ge 1} (a_n \cos nx + b_n \sin nx)$, with arbitrary coefficients. Its partial sums are trigonometric polynomials. Every Fourier series is a trigonometric series, but not every trigonometric series is the Fourier series of an integrable function. The study of these series led Cantor to set theory, through the question of uniqueness of the coefficients.

• ConceptFourier series

  The Fourier series of a periodic function writes it as a sum of sines and cosines, or of complex exponentials $e^{inx}$. Its coefficients are computed by integrals. It turns problems about periodic functions into problems about sequences of coefficients. Whether and in which sense the series converges back to the function is a central question of harmonic analysis. Parseval's identity and Fejér's theorem give two answers.

• ConceptFourier coefficient

  The $n$-th Fourier coefficient $\hat f(n)$ of a $2\pi$-periodic integrable function $f$ measures how much of the wave $e^{inx}$ is present in $f$. It is the average of $f(x) e^{-inx}$ over one period, and it is the inner product of $f$ with the Fourier monomial $e_n$. The sequence $(\hat f(n))_{n \in \Z}$ determines $f$ up to a null set, and the Fourier series of $f$ is $\sum_n \hat f(n) e^{inx}$. The size of the coefficients for large $\abs{n}$ reflects the smoothness of $f$.