# 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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### DFT of a sequence while ignoring some of its elements

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$ with the intention of analyzing its power spectral density by ...
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### Integral, absolute value

Why did we get this identity? $$\frac{1}{\pi} \int_{0}^{2\pi} |x-\pi| dx = \frac{2}{\pi}\int_{0}^{\pi} \pi-x dx$$ And why do we integrate it to: $$-\frac{(\pi-x)^2}{2}$$ Why didn't we just ...
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### What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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### Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
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### How to derive the Fourier Transform of the cosine function?

Given that the function $f(t) = A\cos(\omega t-\phi)$. I cannot get the results for the $f$ domain transform $F(f)$ and the $\omega$ domain transform $F(\omega)$ to be equivalent
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### Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem

This question spanned from a previous interesting one. Let $k$ be a real number greater than $2$ and $$\varphi_k(\xi) = \int_{0}^{+\infty}\cos(\xi x) e^{-x^k}\,dx$$ the Fourier cosine transform of a ...
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### Calculating integral value of Fourier series

Given fourier series: $$\mathrm{S}\left(x\right) = {3 \over \pi}\sum_{n = 0}^{\infty} {\sin\left(\left[2n + 1\right]x\right) \over 2n + 1}\,,\qquad \left\langle -\pi,\pi\right\rangle$$ Evaluate: ...
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### How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
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### Is Fourier series $L^2$?

Let $f\in L^2(0,1)$. I was wondering if the Fourier series of $f$ is a linear map $L^2(0,1)\to L^2(0,1)$. The linearity is obvious, but if $f\in L^2(0,1)$ does $S(f)\in L^2$ or not ? I tried as follow,...
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### Calculate $\int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt$

Let $f\in L^2$. Knowing that $f$ has a Fourier transform given by $\hat{f}(w)=\frac{w}{1+w^4}$ calculate: $$\int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt$$ Im having some trouble in trying to solve ...
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### When is a Fourier transform of a function continuous

I have read about a following claim in a book but cannot prove it. If a function $\psi(x)$ fulfills $\int dx|\psi(x)|^2|x|^2<\infty$, its Fourier transform $\phi(k)$ is continuous and nearly ...
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### Fourier Analysis for Derandomization of Functions

I was wondering if there was an extension to Fourier Analysis on Boolean Functions. Specifically, it's well known that for any boolean function $$f: \{-1,1\}^{n} \rightarrow [-1,1]$$ we can decompose ...