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The Fourier series of a complex-valued P-periodic function (), integrable over the interval [,] on the real line, is defined as a trigonometric series of the form =, such that the Fourier coefficients are complex numbers defined by the integral [15] [16] = .
If () is a periodic function, with period , that has a convergent Fourier series, then: ^ = = (), where are the Fourier series coefficients of , and is the Dirac delta function. In other words, the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.
However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere. It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by [9]
Fourier series. When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
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.
For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.
A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation, [20] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series ...