Very interesting. The integral [tex] \int^\infty_{-\infty}e^{2\pi ikx} dx [/tex] does in some ways behave like a delta function. And the delta function is an ideal function. However it's own Fourier transform is an exponential, which is a real funtion. The Fourier transform as an operator on Hilbert space is unitary, and squares to -1. Neither the delta function nor the exponential function are in Hilbert space, the latter because it doesn't satisfy boundary conditions, and the former because it isn't even a funtion. The idea is very informal and lacks rigour. I have never seen it given a rigorous basis.HallsofIvy said:Suppose you were to ask for the Fourier Series for f(x)= cos(x)?
Since the Fourier Series is, by definition, a sum of sines and cosines that add to f(x).
Since f(x)= cos(x), its Fourier series coefficients are just a1= 1, all other coefficients are 0. The delta "function" (it's really a "distribution" or "generalized function") is the functional version of that.
The Fourier transform of exp(ix) is a delta function because it represents a pure sinusoidal wave with a frequency of x. This means that all other frequencies are zero, resulting in a single spike at the frequency x in the frequency domain, which is the definition of a delta function.
The Fourier transform of exp(ix) demonstrates the properties of the Fourier transform, such as linearity, time and frequency shifting, and convolution. It also shows the duality between the time and frequency domains, as the delta function in the frequency domain corresponds to a pure sinusoidal wave in the time domain.
The Fourier transform of exp(ix) is essential in signal processing because it allows us to decompose signals into their constituent frequencies. This is useful in analyzing signals, filtering out unwanted frequencies, and designing communication systems.
Yes, the Fourier transform of exp(ix) is a special case of the Fourier transform and can be generalized to other functions. It is a fundamental tool in mathematics and has various applications in physics, engineering, and other fields.
The Fourier transform of exp(ix) is orthogonal to all other functions in the frequency domain, except for multiples of itself. This is because it represents a pure sinusoidal wave with a single frequency, which is orthogonal to all other frequencies. This property is crucial in the analysis of signals and systems, where orthogonality is used to simplify calculations and find unique solutions.