With the following Mathematica program: Clear[x, n, nn] nn = Prime[13] + 1; A = -Sum[Re[x^N[ZetaZero[n]]], {n, 1, 50}]; Plot[A, {x, 0, nn}, PlotRange -> {-80, 120}] I plotted the function: $$f(x)=-\sum _{n=1}^{50} \Re\left(x^{\rho _n}\right)$$ Where $\rho _n$ is the $n$-th Riemann zeta zero. With this second program: Clear[n, k, x, nn] nn = Prime[13] […]

My teacher gave us this version of Riemann–Lebesgue lemma in class: Let $g(t)$ be an absolutely integrable function on $[a,b]$, then $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt=0$$ Similarly for $\cos(px)$ Then, he told us to look for an example of a function $g(t)$ such that it is integrable (in the improper sense) such that $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt\neq0$$ i.e., […]

Show that the Fourier series for the square wave function $$f(t)=\begin{cases}-1 & -\frac{T}{2}\leq t \lt 0, \\ +1 & \ \ \ \ 0 \leq t \lt \frac{T}{2}\end{cases}$$ is $$f(t)=\frac{4}{\pi}\left(\sin\left(\frac{2\pi t}{T}\right)+\frac{\sin(\frac{6\pi t}{T})}{3}+\frac{\sin(\frac{10\pi t}{T})}{5}+……\right)$$ I understand that the general Fourier series expansion of the function $f(t)$ is given by $$f(t)=\frac{a_0}{2}+ \sum_{r=1}^{r=\infty}\left(a_r\cos\left(\frac{2\pi r t}{T}\right)+b_r\sin\left(\frac{2\pi r t}{T}\right)\right)$$ But […]

I’m currently attempting to solve the following problem: Given the function $f$ defined on the interval $(0, \pi)$ by $f(x) = \cos{2x}$, find the $2\pi$-periodic, even extension of $f$ and compute the cosine Fourier series of $f$. However, I seem to be having difficulty obtaining a reasonable result for this problem. Here’s what I tried. […]

Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a torus, which is homeomorphic to the Cartesian product of two circles: $S^1\times S^1$, the Fourier series should be defined by: $$f(\phi,\theta)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}F(n,m)e^{im\phi}e^{in\phi},$$ where $$F(n,m)=\frac{1}{2\pi}\int_0^{2\pi}\int_0^{2\pi}f(\phi,\theta)e^{-im\phi}e^{-in\phi}~~r(R+r\sin(\phi))d\theta […]

Let $f$ be a continuous and integrable function with period $2\pi$. Consider its fourier coefficients with respect to the orthonormal system $\{ \frac {1}{\sqrt{2\pi} } e^{inx}\}$. If all the Fourier coefficients are zero, prove that $f$ is the zero function. I think it is a very natural proposition but I find myself stuck because we […]

So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this. I know ‘big ell 2’ and ‘little el 2’ are vector spaces. However I need a little bit more understanding on what these are. The lecture […]

Ok so I want to prove the above expression, I substituted the complex fourier series for f and using the fact f may be complex-valued, carried on by representing $|f(x)|^2$ as $f(x)f(x)^\ast$ where the ast represents complex conjugation. The problem is when now I substitute the complex fourier series I get three summations, one is […]

I’m considering a proof of the convergence of the Fourier series. It begins by considering the full Fourier series of the periodic extension of $\phi$ defined on $[-\ell, \ell]$. The full Fourier series is $$ \sum_{n=-\infty}^\infty C_ne^{\frac{in\pi x}\ell}, \quad C_n = \frac{1}{2\ell}\int_{-\ell}^\ell \phi(x)e^{\frac{-in\pi x}{\ell}}dx. $$ Then we attempt to bound $|C_n|$ as $$ |C_n| \leq […]

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in \ell^2(\mathbb{Z}^n)$ where the above sums converge? My ideas so far: $ij \leq i^2 + j^2$ shows that $C=1$ works. Is 1 the sharpest possible bound? (This comes from showing that for $u,f$ periodic and […]

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