# Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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### Fractional Fourier Transform of $\sqrt{c} x( c(t - \tau))$

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
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### Solving the discrete Fourier Transform of $\sin(x)+\sin(2x)$

I'm new to the discrete Fourier Transform and have managed to do a few examples by hand, (for example the Fourier Transform of $\sin(x)$). I now want to try the transform on signals that are made up ...
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### Counstruct a sequence of Schwartz functions

Here is an exercise from Wolff's lectures on harmonic analysis, p26: Using translation and multiplication by characters, construct a sequence of Schwartz functions $\lbrace\phi_n\rbrace$ so that Each ...
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### Expression $f$ on $\mathbb{T^n}$ in terms of $\hat{f}$.

I'm studying Real analysis, Folland section 8.4 page 257. I can't undersfand the below highlighted statement. If $f\in L^1(\mathbb{T^n})$ and $\hat{f}$ $\in l^1(\mathbb{Z^n})$, then the Fourier series ...
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### How can you derive the spacetime Fourier transform of the free Schrodinger evolution rigorously?

I'm trying to compute the spacetime Fourier transform of the free Schrodinger evolution. Consider $f\in L^2(\mathbb{R}^d)$ and $e^{it\Delta}f=:\mathcal{F}^{-1}(e^{-it|\xi|^2}\hat{f}(\xi))$ its free ...
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### Formula connecting Fourier transforms of function and its derivative

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then $$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$ (Here $\mathcal{F}$ represents ...
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### Cesaro summation of the inverse Fourier transform.

Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity ...
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