# 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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### Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $f$ be a function such that $f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R})$ (f is $2\pi$-periodic) such that $\forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
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### The Fourier series $\sum_{n=1}^\infty (1/n)\cos nx$

The series $$\sum_{n=1}^\infty \frac{\sin nx}{n}$$ is the Fourier series of the odd $2\pi$-periodic extension of $(\pi-x)/2, 0<x<\pi$. My question is : $$\sum_{n=1}^\infty \frac{\cos nx}{n}$$ ...
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### Conceptual/Graphical understanding of the Fourier Series.

I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
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### Evaluate $\sum^\infty_{n=1} \frac{1}{n^4}$using Parseval's theorem (Fourier series)

Evaluate $\sum^\infty_{n=1} \frac{1}{n^4}$using Parseval's theorem (Fourier series). I have , somehow, to find the sum of $\sum_{n=1}^\infty \frac{1}{n^4}$ using Parseval's theorem. I tried some ...
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### Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
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### How does knowing a function as even or odd help in integration ??

So, I am learning Fourier Series and it involves integration. I am not too good at integration. Now, the resource I use is videos by Dr. Chris Tisdell. In the ...
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### Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
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### If square waves are square integrable, why doesn't fourier expanding work?

If square waves are square integrable, then why does expanding on a fourier basis not recover the equation?
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### Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
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### Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = x^2$...
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### Can any continous,bounded function have a fourier series?

In particular,can an oscillatory function with some decay term ( i.e e^(-t)*cos(kt) have a fourier series representation? All the articles I read said that the function has to be periodic,but this one ...
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### Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem. If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$. How to prove it?
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We are all aware of the dilogarithm function (Spence's function): $$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$ Also it is known that: $$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ... 1answer 269 views ### An integral that might be related to the modified Bessel function of second kind It is known that the modified Bessel Function K_z(a) (a>0)can be expressed as a Fourier transform$$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$Can ... 1answer 280 views ### Fourier series of \sqrt{1 - k^2 \sin^2{t}} I'm struggling with a Fourier series. I need to find the Fourier series of the following function. That's the function under study: f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]. The function ... 4answers 493 views ### Evaluate  \sum_{n=1}^{\infty} \frac{\sin \ n}{ n }  using the fourier series I am a beginner with Fourier series and I have to evaluate the sum$$\sum_{n =1}^{\infty}{\sin\left(n\right) \over n}$$I don't know which function I have to take to evaluate the fourier series ...... 1answer 743 views ### Roots of a finite Fourier series? In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ... 1answer 1k views ### A function and its Fourier transform cannot both be compactly supported I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know f is a trigonometric polynomial, I see how to finish the ... 1answer 528 views ### Series expansion of \coth x using the Fourier transform Hi I have research about the series of coth but all of the solutions emerges from integral on a contour, Could you calculate the fourier transform of coth? Is that possible at all?My goal is to reach :... 1answer 892 views ### Absolute convergence of Fourier series of a Hölder continuous function Suppose that f is 2 \pi periodic and Hölder continuous of order \alpha > 1/2. Show that the Fourier series of f converges absolutely. So we know that f(x+2 \pi t) = f(x) for all t \in \... 0answers 51 views ### Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function. NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ... 1answer 81 views ### Riemann Lebesgue Lemma Clarification If f is continuous real-valued function, does the Riemann Lebesgue Lemma give us that \int_{m}^k f(x) e^{-inx}\,dx \rightarrow 0\text{ as } n\rightarrow \infty for all m\le k? Specifically, is ... 1answer 65 views ### Prove that the function \xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ is C^{\infty} [closed] Prove that the following function is C^\infty in the point \xi=0:$$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ Any ideas how to prove this? I am trying to think ...
Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$. I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
### Convergence of series of functions: $f_n(x)=u_n\sin(nx)$
Let $f_n(x)=u_n\sin(nx)$ where $\displaystyle\sum f_n$ converges pointwise, and $\displaystyle x \mapsto \sum_{n=0}^{+\infty} f_n(x)$ is continuous. Prove that $u_n\rightarrow 0$ when n tends ...