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Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function.
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain , the bilinear form may be the integral of the product of functions over the interval:
A closely related topic is the approximation of functions by generalized Fourier series, ... of a series of terms based upon orthogonal ... or calculator (e.g ...
A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion is applied to periodic functions. In contrast, a generalized Fourier series uses any set of ...
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. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes).
The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions. [2] Walsh functions, the Walsh system, the Walsh series, [3] and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh.
The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions. Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be , = ()
The Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials. A rational Legendre function of degree n is defined as: R n ( x ) = 2 x + 1 P n ( x − 1 x + 1 ) . {\displaystyle R_{n}(x)={\frac {\sqrt {2}}{x+1}}\,P_{n}\left({\frac {x-1}{x+1 ...