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...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

''for the convergence of Fourier series'' [[Category:Harmonic analysis on Euclidean spaces]]

...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:

...$ 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 $t'$.

''for the convergence of Fourier series'' [[Category:Harmonic analysis on Euclidean spaces]]

...under Fourier transformation (cf. [[Fourier transform|Fourier transform]]) on the space $ L _ {2} ( X) $: dimensional Euclidean space and $ \mu ( x) $

$#C+1 = 44 : ~/encyclopedia/old_files/data/C025/C.0205040 Conjugate harmonic functions, A pair of real harmonic functions $ u $

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

in the Euclidean space $ \mathbf R ^ {n} $, is a given continuous function on the sphere $ S _ {n} ( 0 , R ) $

..., \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

...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

on which $ G $ on $ M $

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 $\Omega$:

...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

depending on a parameter $ \lambda \in \Lambda $; is an orthogonal random measure on $ \Lambda $(

...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

...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

$#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

on $ [ a, b] $ on the circle $ | \lambda | = R $.