Decompositions of inner product spaces into orthonormal bases
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In mathematics, a generalized Fourier series expands a square-integrable function defined on an interval over the real line. The constituent functions in the series expansion form an orthonormal basis of an inner product space. While a Fourier series expansion consists of only trigonometric functions, a generalized Fourier series is a decomposition involving any set of functions that satisfy the Sturm-Liouville eigenvalue problem. These expansions find common use in interpolation theory.[1] It is expressed by a series of sinusoids that can be stated in various forms. In essence, let us consider a pair of functions, where t is a variable (usually time), and m, n are real multipliers of t, reflecting the length of interval.
^Howell, Kenneth B. (2001-05-18). Principles of Fourier Analysis. Boca Raton: CRC Press. doi:10.1201/9781420036909. ISBN 978-0-429-12941-4.
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a generalizedFourierseries expands a square-integrable function defined on an interval over the real line. The constituent functions in the series expansion...
A Fourierseries (/ˈfʊrieɪ, -iər/) is an expansion of a periodic function into a sum of trigonometric functions. The Fourierseries is an example of a...
simpler trigonometric functions. Fourier analysis grew from the study of Fourierseries, and is named after Joseph Fourier, who showed that representing...
Fourier may refer to: Fourier (surname), French surname Fourierseries, a weighted sum of sinusoids having a common period, the result of Fourier analysis...
field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. In this article, f denotes...
requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space,...
In mathematics, the question of whether the Fourierseries of a periodic function converges to a given function is researched by a field known as classical...
in the continuous case. Thus they can also be thought of as a generalizedFourierseries in which the basis functions are the normal modes of an atmosphere...
is found by using the Fourier transform for functions on the real line or by Fourierseries for periodic functions. Generalizing these transforms to other...
the approximation of functions by generalizedFourierseries, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials...
finite-dimensional subspaces cannot serve as a basis.) Markushevich basis GeneralizedFourierseries Orthogonal polynomials Haar wavelet Banach space see Schauder...
{1}{a}}\rho _{m,n}.} The general solution A then takes the form of a generalizedFourierseries of terms involving products of Jn(km,nr) and the sine (or cosine)...
quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transform...
orthogonal solution functions (a.k.a. eigenfunctions), leading to generalizedFourierseries. Eigenvalues and eigenvectors Hilbert space Karhunen–Loève theorem...
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex...
Polynomial sequences of binomial type Biorthogonal polynomials GeneralizedFourierseries Secondary measure Sheffer sequence Sturm–Liouville theory Umbral...
infinite version of a trigonometric polynomial. A trigonometric series is called the Fourierseries of the integrable function f {\textstyle f} if the coefficients...
transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourierseries of an integrable function. These were disconnected...
A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis...
fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform...
the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform...
founder of Steklov Institute of Mathematics, proved theorems on generalizedFourierseries Bella Subbotovskaya, specialist in Boolean functions, founder...
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