Notes on Dirichlet Conditions in Fourier Transformation
In mathematics, the Dirichlet conditions are under Fourier Transformation are used in order to valid condition for real-valued and periodic function f(x) that are being equal to the sum of Fourier series at each point (where f is a continuous function). The behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). Following post provide Signal System Notes on Drichlet’s Condition in Fourier Transformation to help students in order to understand this topic.
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Dirichlet Conditions in Fourier Transformation are as follows:
Dirichlet Conditions
- f(x) must absolutely integrable over a period.
- f(x) must have a finite number of exterma in any given interval, i.e. there must be a finite number of maxima and minima in the interval.
- f(x) must have a finite number of discontinues in any given interval, however the discontinuity cannot be infinite.
- f(x) must be bounded.
The last three conditions are satisfied if f is a function of bounded variation over a period.
Dirichlet’s Theorem for 1-Dimensional Fourier Series
We state Dirichlet’s theorem assuming f is a periodic function of period 2π with Fourier series expansion where
The analogous statement holds irrespective of what the period of f is, or which version of the Fourier expansion is chosen.
Dirichlet’s Theorem
If f satisfies Dirichlet conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by
- where the notation
A function satisfying Dirichlet’s conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where f is continuous signal function
Thus Dirichlet’s theorem says in particular that the Fourier series for f converges and is equal to f wherever f is continuous.
Solution of Scalar Wave Equation
Solution of scalar wave equation in parabolic approximation will be only consider through their finite-difference and Fourier-transform techniques. A comparison of numerical results obtained by the two methods is presented, and a comparison with other analytically or numerical methods is also given. In the numerous cases studied it is shown that the finite-difference method yields a large, order-of-magnitude range improvement in accuracy or computational speed when compared with the Fourier-transform method.
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