Tagged Questions

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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Derivative of Fourier transform with respect to intermediate variable

I am studying a system with a characteristic, say $\zeta$, that varies in 3D real space. I can use this characteristic to calculate the value of a second characteristic $\beta$. In other words, I have ...
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Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
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If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
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Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$. $f(w)$ is a Schwartz function. This is a part of the proof of Fourier inversion ...
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Integration on $R^d$ about Changing Coordinate

I have the formula $$\int_{\mathbb R^d} F(x)dx=\int_{S^{d-1}}\int_0^\infty F(r\gamma)r^{d-1}drd\sigma(\gamma)$$ Problem: Apply this to $F(x)=g(r)f(\gamma)$, where $x=r\gamma$, to prove that for ...
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