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Fourier transforms. A Fourier series ( / ˈfʊrieɪ, - iər / [ 1]) 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. [ 2] By expressing a function as a sum of sines and cosines, many problems involving ...
In general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. If f is of bounded variation, then its Fourier series converges everywhere. If f is continuous and its Fourier coefficients are absolutely summable, then the ...
The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions f , ∑ n f ^ ( n ) = ∑ n f ( n ) . {\displaystyle \sum _{n}{\hat {f}}(n ...
Fourier transforms. In mathematics, Fourier analysis ( / ˈfʊrieɪ, - iər /) [ 1] is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum ...
If f is an odd function with period , then the Fourier Half Range sine series of f is defined to be = = which is just a form of complete Fourier series with the only difference that and are zero, and the series is defined for half of the interval.
Parseval's identity. 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 ...
Poisson summation formula. In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original ...
Dirichlet–Jordan test. In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real -valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the ...