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and the integral of the product of the two sine functions vanishes. [1] Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.
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 ...
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.
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.
Periodic function; Trigonometric function; Trigonometric polynomial. Exponential sum; Dirichlet kernel; Fejér kernel; Gibbs phenomenon; Parseval's identity; Parseval's theorem; Weyl differintegral; Generalized Fourier series. Orthogonal functions; Orthogonal polynomials; Empirical orthogonal functions; Set of uniqueness
A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials.
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 , = ()
Any sufficiently smooth real-valued phase field over the unit disk (,) can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with the Fourier series. We have