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1. Fourier series: A Fourier series (/ˈfʊrieɪ, -iər/) 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. (Decomposition of periodic functions into sums of simpler sinusoidal forms) [100%] 2024-01-03 [Fourier series] [Joseph Fourier]...
2. Fourier series: In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), refers to an infinite series representation of a periodic function ƒ of a real variable ξ, of period P: In the case of a complex-valued function ƒ(ξ), Fourier's theorem states ... [100%] 2024-01-04
3. Fourier series: Fourier series express a piecewise continuous, bounded, periodic function as a linear combination of orthogonal sine and cosine functions. The seeds of the modern theory were developed by Joseph Fourier. [100%] 2023-02-28 [Calculus]
4. Fourier series: of a function $f$ in a system of functions $\{\phi_n\}$ which are orthonormal on an interval $(a,b)$. The series $$ \sum_{k=0}^\infty c_k \phi_k $$ whose coefficients are determined by \begin{equation} \label{eq1} c_k = \int_a^b f(x ... (Mathematics) [100%] 2024-01-04
5. Generalized Fourier series: In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. (Decompositions of inner product spaces into orthonormal bases) [81%] 2024-01-04 [Fourier analysis]
6. Discrete Fourier series: In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. [81%] 2023-12-06 [Fourier analysis]
7. Generalized Fourier series: In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. (Decompositions of inner product spaces into orthonormal bases) [81%] 2024-01-03 [Fourier analysis]
8. Summation of Fourier series: The construction of averages of Fourier series using summation methods. The best developed theory of the summation of Fourier series is that which uses the trigonometric system. (Mathematics) [70%] 2023-10-18
9. Fourier series in orthogonal polynomials: A series of the form $$\sum_{n=0}^\infty a_nP_n\label{1}\tag{1}$$ where the polynomials $\{P_n\}$ are orthonormal on an interval $(a,b)$ with weight function $h$ (see Orthogonal polynomials) and the coefficients $\{a_n\}$ are calculated from the ... (Mathematics) [63%] 2023-10-12
10. Fourier series of an almost-periodic function: A series of the form $$f(x)\sim\sum_na_ne^{i\lambda_nx},\label{*}\tag{*}$$ where the $\lambda_n$ are the Fourier indices, and the $a_n$ are the Fourier coefficients of the almost-periodic function $f$ (cf. Fourier indices of an almost-periodic ... (Mathematics) [57%] 2023-10-18
11. Fourier-Stieltjes series: A series $$ { \frac{a _ {0} }{2} } + \sum _ {n = 1 } ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx), $$ where for $ n = 0, 1 \dots $ $$ a _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi ... (Mathematics) [81%] 2024-01-04
12. Fourier-Bessel series: The expansion of a function $ f $ in a series $$ \tag{* } f ( x) = \ \sum _ {m = 1 } ^ \infty c _ {m} J _ \nu \left ( x _ {m} ^ {( \nu ) } \cdot { \frac{x}{a} } \right ) ,\ \ 0 - 1/2 $( cf. (Mathematics) [81%] 2023-10-22
13. Fourier-Haar series: Haar–Fourier series Consider an interval $(a,b)$, a measure $\mu$ on it and a corresponding complete orthonormal system of functions $\phi_0,\phi_1$ (so that $\int_a^b\phi_k(x)\phi_l(x)\,d\mu(x)=\delta_{kl}$). The Fourier series of ... (Mathematics) [81%] 2024-01-04