# Tagged Questions

A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calcalating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a ...

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### Having trouble understanding the proof for Fourier Transform Scaling Property [closed]

Starting from Plancherel's Theorem: $$f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(k)e^{ikx}dk ...(1)$$ $$F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx ...(2)$$ I need to ...
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### Characteristics of an Optical System in the Fourier Domain

An imaging system can be characterized by its point spread function (PSF), which in most cases is space-variant. The final image is the result of the convolution of the PSF with the object (2d ...
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### A question about Fourier Transformation [closed]

Recently I try to evaluate a integral in a paper: $$\Gamma(x,y)=\int_{-\infty}^{\infty} \frac{dk}{2\pi} \sqrt{k^2+m^2} e^{ikx}$$ This is the Fourier Transform of: $$f(k)=\sqrt{k^2+m^2}$$ The ...
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### Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
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### Estimation of the autocorrelation for data on finite size interval

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
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### Units of a fourier transform end energy density of the time dependent force

I have a signal, which is time dependent force F(t) (obtained from the Atomic Force Microscope) I wanted to estimate the energy content in the signal for given frequency band. I have calculated Energy ...
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### Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 \end{...
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### Why can you only measure velocity or location in a particle?

I was talking to a family friend in the field of optics at a quantum scale (not sure the proper name for this) and he was explaining to me why you can only determine either the velocity or location of ...
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### What is the significance of the Fourier coefficients?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...