The earlier publish confirmed how we are able to take the Fourier rework of capabilities that don’t have a Fourier rework within the classical sense.
The classical definition of the Fourier rework of a operate f requires the integral of |f| over the actual line to be finite. This suggests f(x) should method zero as x goes to ∞ and −∞. A relentless operate received’t do, and but we received round that within the earlier publish. Distribution principle even permits you to take the Fourier rework of capabilities that develop as their arguments go off to infinity, so long as they don’t develop too quick, i.e. like a polynomial however not like an exponential.
On this publish we wish to take the Fourier rework of capabilities like sine and cosine. In the event you learn that sentence as saying Fourier sequence, you may have the suitable intuition for classical evaluation: you are taking the Fourier sequence of periodic capabilities, not the Fourier rework. However with distribution principle you can take the Fourier rework, unifying Fourier sequence and Fourier transforms.
For this publish I’ll be defining the classical Fourier rework utilizing the conference
and generalizing this definition to distributions as within the earlier publish.
With this conference, the Fourier rework of 1 is δ, and the Fourier rework of δ is 2π.
One can present that the Fourier rework of a cosine is a sum of delta capabilities, and the Fourier rework of a sine is a distinction of delta capabilities.
It follows that the Fourier rework of a Fourier sequence is a sum of delta capabilities shifted by integers. In reality, in the event you convert the Fourier sequence to advanced type, the coefficients of the deltas are precisely the Fourier sequence coefficients.
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