* {{MSCwiki|28}} [[:Category:Measure and integration]] {For analysis on manifolds, see {{MSCwiki|58-XX}}}
* {{MSCwiki|30}} [[:Category:Functions of a complex variable]] {For analysis on manifolds, see {{MSCwiki|58-XX}}}
208 members (14 subcategories, 0 files) - 22:24, 28 January 2012
...of the Fourier series (cf. [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]) corresponding to the given almost-
The coefficients $a_n$ are completely determined by the theorem on the existence of the mean value
1,012 bytes (164 words) - 21:53, 7 November 2014
''for the convergence of Fourier series''
[[Category:Harmonic analysis on Euclidean spaces]]
1 KB (205 words) - 12:30, 27 September 2012
...tegral transform|Integral transform]]). It is a linear operator $F$ acting on a space whose elements are functions $f$ of $n$ real variables. The smalles
...rse mapping $F^{-1}$ (the inverse Fourier transform) is the inverse of the Fourier transform and is given by the formula:
6 KB (971 words) - 12:28, 1 February 2020
...$ in $n$ variables $x=(x_1,\dots,x_n)$ to the unit sphere $S^{n-1}$ of the Euclidean space $E^n$, $n\geq3$. In particular, when $n=3$, the spherical harmonics a
...0$, a zonal spherical harmonic $Z_{t'}^{(k)}(x')$ exists which is constant on any parallel of the sphere $S^{n-1}$ that is orthogonal to the vector $t'$.
3 KB (519 words) - 19:48, 22 December 2018
''for the convergence of Fourier series''
[[Category:Harmonic analysis on Euclidean spaces]]
2 KB (342 words) - 12:49, 6 October 2012
...under Fourier transformation (cf. [[Fourier transform|Fourier transform]]) on the space $ L _ {2} ( X) $:
dimensional Euclidean space and $ \mu ( x) $
6 KB (796 words) - 08:06, 6 June 2020
$#C+1 = 44 : ~/encyclopedia/old_files/data/C025/C.0205040 Conjugate harmonic functions,
A pair of real harmonic functions $ u $
4 KB (617 words) - 17:46, 4 June 2020
A partial sum of the [[Fourier series|Fourier series]] of a given [[Function|function]].
In the classical one-dimensional case where a function $f$ is integrable on the segment $[-\pi,\pi]$ and
6 KB (1,047 words) - 11:37, 13 February 2024
in the Euclidean space $ \mathbf R ^ {n} $,
is a given continuous function on the sphere $ S _ {n} ( 0 , R ) $
11 KB (1,592 words) - 12:54, 31 January 2022
..., \dots$, $2 \pi$-periodic in each variable. Consider its [[Fourier series|Fourier series]] $\sum _ { k } \hat { f } ( k ) e ^ { i k x }$, where $x = ( x _ {
...er coefficients exists, thus the definition of a multi-dimensional partial Fourier sum presents many problems and points of interest intimately connected to g
17 KB (2,613 words) - 19:24, 12 February 2024
...u$ invariant. For $x \in X$ and $g \in G$, $g.x$ denotes the action of $g$ on $X$. A family $\mathcal{K}$ of compact subsets of $X$ is said to have the P
...denote the characteristic function of $K$ and $\widehat { \chi }_{K}$ its Fourier transform, which is an entire function of [[Function of exponential type|ex
9 KB (1,368 words) - 20:29, 10 January 2021
on which $ G $
on $ M $
20 KB (2,996 words) - 08:42, 16 December 2019
...s space for $k$, namely, the set $M = G / H$ of left cosets of $H$ in $k$, on which $k$ acts by the formula
..., differentiable). If a Lie group $k$ acts transitively and differentiably on a differentiable manifold $N$, then, for any point $x _ { 0 } \in M$, the s
20 KB (3,056 words) - 13:44, 17 October 2019
By the analysis of E. Wigner [[#References|[a2]]], [[#References|[a3]]], all states which d
...mathcal{F} ( f _ { l } ) )$, one defines asymptotic fields by their action on the vacuum vector $\Omega$:
9 KB (1,385 words) - 20:57, 8 February 2024
...4.5). For $n=2$ it coincides with the real-analytic [[Eisenstein series]] on the upper half-plane (cf. [[Modular form]]; [[#References|[a5]]], V.Sect. 5
...\mathbf{Z}\lambda_1 + \cdots + \mathbf{Z}\lambda_n$ in an $n$-dimensional Euclidean vector space $(V,\sigma)$. One has
5 KB (734 words) - 20:12, 5 October 2017
depending on a parameter $ \lambda \in \Lambda $;
is an orthogonal random measure on $ \Lambda $(
17 KB (2,406 words) - 20:00, 12 January 2024
...rphic forms (cf. also [[Automorphic form|Automorphic form]]) involve their Fourier coefficients. Here, the special case of holomorphic modular forms $f$ of we
...plane. Therefore it has period $1$ in the real part of $z$ and must have a Fourier expansion
13 KB (1,907 words) - 07:36, 22 March 2023
...a function $f$ in a system of functions $\{\phi_n\}$ which are orthonormal on an interval $(a,b)$.''
...efficients of $f$. In general it is assumed that $f$ is square integrable on $(a,b) $. For many systems $\{\phi_k\}$ this requirement can be relaxed by
21 KB (3,473 words) - 19:50, 26 April 2012
$#C+1 = 165 : ~/encyclopedia/old_files/data/W097/W.0907760 White noise analysis
...point [[#References|[a7]]] — is that of an infinite system of coordinates on which to base an infinite-dimensional calculus. More precisely, the startin
27 KB (3,916 words) - 19:20, 13 January 2024