Category: Fourier series

Sawtooth wave
The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an inter
Trigonometric series
In mathematics, a trigonometric series is a infinite series of the form an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function if the terms and h
Fejér's theorem
In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: Fejér's Theorem — Let be a continuous function with period , let be the nth partial sum of the F
Fourier–Bessel series
In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions. Fourier–Bessel series are used i
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Henry Wilbraham and rediscovered by J. Willard Gibbs, is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable p
Triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave cont
Beurling algebra
In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling, usually it is an algebra of periodic functions with Fourier series ExampleWe may consider the alge
Wiener algebra
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series. Here T denotes the circle group.
Parseval's identity
In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pyt
Dini–Lipschitz criterion
In mathematics, the Dini–Lipschitz criterion is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Ulisse Dini, as a s
Half range Fourier series
In mathematics, a half range Fourier series is a Fourier series defined on an interval instead of the more common , with the implication that the analyzed function should be extended to as either an e
F. and M. Riesz theorem
In mathematics, the F. and M. Riesz theorem is a result of the brothers Frigyes Riesz and Marcel Riesz, on analytic measures. It states that for a measure μ on the circle, any part of μ that is not ab
Fourier series
A Fourier series (/ˈfʊrieɪ, -iər/) is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose function
Riesz–Fischer theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was prove
Fejér kernel
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is na
Summability kernel
In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are parti
Denjoy–Luzin theorem
In mathematics, the Denjoy–Luzin theorem, introduced independently by Denjoy and Luzin states that if a trigonometric series converges absolutely on a set of positive measure, then the sum of its coef
Fourier sine and cosine series
In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.
Dini criterion
In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Ulisse Dini.
Square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In
Finite Fourier transform
In mathematics the finite Fourier transform may refer to either * another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., (pp. 52–53) describes the finite Fourier tra
Sigma approximation
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities. A σ-approximated summation for a series of period T
Trigonometric Series
Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published i
Dirichlet conditions
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavi
Conjugate Fourier series
In the mathematical field of Fourier analysis, the conjugate Fourier series arises by realizing the Fourier series formally as the boundary values of the real part of a holomorphic function on the uni
Dini test
In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Uliss
Pinsky phenomenon
In mathematics, the Pinsky phenomenon is a result in Fourier analysis. This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier tran
Weyl integral
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0 and a Fourier series. In
Dirichlet kernel
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of functions defined as where n is any nonnegative integer. The k
Fourier amplitude sensitivity testing
Fourier amplitude sensitivity testing (FAST) is a variance-based global sensitivity analysis method. The sensitivity value is defined based on conditional variances which indicate the individual or jo
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathemat