# Tagged Questions

Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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### Can a well-behaved, positive-definite function $\phi(x)$ always be represented by $\phi(x)=\sum_{n=0}^{\infty}\mid a_{n}\psi(x)_n\mid^{2}$?

It's well known that some $\phi(x^{\mu})$ fulfilling a basic set of criteria (on a manifold M of arbitrary dimension lets say dim=1 here) may be represented as a fourier series of orthogonal ...
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### Fourier series Coefficients and wolframalpha

1) Please can my answers be checked, including my final Fourier series. 2) Is it possible to use Wolframalpha to check my answers? If so, how will I go about doing this? Deduce the Fourier series ...
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### Fourier Transform: Musical instruments

How do I Fourier Analyse the music produced by a musical instrument? What I mean is that what tools/applications are best suited to Fourier Analyse waves from musical instruments?
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### How does shifting make this function odd?

[Supply current of 3 phase semi converter] I've been told that shifting this waveform left by 30 + a/2 will make it odd. Odd means f(a) = -f(a), right? So, how it that happening here? Or am i ...
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### Proving that the limit of an integral of a series exists

The goal is to show that the following limit exists $$\lim_{T\to\infty} \frac{1}{T}\int_{-T}^T f(x)dx$$ where $$f(x)=\sum_{n=1}^\infty \frac{e^{ia_n x}}{n^2}$$ I already showed that $f$ is bounded ...
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### Fourier series from wikipedia

How did this page come up with the Fourier coefficients? It basically jump after coming up with this $$A_n=\sqrt{a^2_n+b^2_n} , \quad\phi_n=\arctan\left(\frac{a_n}{b_n}\right).$$
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### Ways to justify this interchange of summation and integration

In evaluating this integral: $$\int_0^\infty \frac{\Im{\left(e^{e^{ix}} \right)}}{x}\text{d}x$$ My means of evaluation was to expand the numerator of the integrand as a fourier series (a.k.a. Taylor ...
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### Convergence of a cosine series

Given that $$\sum_{n=1}^\infty a_n<\infty,$$ and that $$\lim_{n\to \infty}b_n=0$$ Is the series $$\sum_{n=0}^\infty a_nb_n^{-2}(1-cos(b_n))$$ necessarily convergent?
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### Fourier Series of this function

Find the Fourier series of this function, only by using sine functions. This is not a homework, I'm just practicing different problems for an exam. I know that all coefficients, except b0 should be 0. ...
It's proven that $\sum_{k=1}^{n}\sin(k\theta)/k$ is uniformly bounded for all $\theta\in\mathbb{R}$ and all $n\geq 1$. So there exists a $M>0$ such that $$\left|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k}... 1answer 26 views ### Fourier series: Where is the source of resonance in the original input signal? I understand that Fourier series approximate the input signal well and series converge to the original function. But, the system "ODE", such as "x''+Ax'+Bx=f(t)", where f(t) is a periodic function ... 2answers 42 views ### Complex series should sum to zero but it's a puzzle If we have a finite sum defined as$$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$(where k is an integer and N is divisible by 4), then how can we show that this sum is equal to ... 1answer 31 views ### How does the orthogonality of sine and cosine figure in the Fourier series? Okay so this is my first time asking a question so if I've made a mistake pls inform context: I'm in high school and I'm going to use the Fourier Series for easily doable applications, that isn't the ... 0answers 14 views ### Fourier coefficients of \;\log\log I was curious if there is an effective way to compute (the asymptotic of) the Fourier coefficients of$$ F(x)= \log\log\left(\frac{1}{\left\lvert x\right\rvert}\right) \cdot \chi\left(\left\lvert x\...
I'm doing "Digital Image" online course. I tried to solve the following question $x(n_1,n_2)$ is defined as $x(n_1,n_2)=(−1)^{(n_1+n_2)}$ when $0≤n_1$, $n_2≤2$ and zero elsewhere. Denote by \$X(k_1,k_2)...