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Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, L p spaces, summability methods and the Cesàro mean.
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity . It is named after the Hungarian mathematician Lipót Fejér (1880–1959).
The convolution of D n (x) with any function f of period 2 π is the nth-degree Fourier series approximation to f, i.e., we have () = () = = ^ (), where ^ = is the k th Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function 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 discontinuity).
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula () =, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation .
Proof: a) Given that is the mean of , the integral of which is 1, by linearity, the integral of is also equal to 1.. b) As () is a geometric sum, we get an simple formula for () and then for (),using De Moivre's formula :
By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "f(x) is smooth if and only if f̂(ξ) quickly falls to 0 for | ξ | → ∞."
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 function's Fourier transform.