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Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$\left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ... 0answers 37 views ### Invariant functions under integral transforms We all know Fourier transform has invariants such as e^{-x^2}, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ... 2answers 62 views ### How to estimate (compute) Fourier transform? Let f:\mathbb R \to \mathbb R such that$$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$for x\in \mathbb R - \{ 0, -1, 1 \} and f(x):= \pi  for x=0 and f(x)=-\frac{\pi}{2} for x= -1, 1. ... 2answers 42 views ### for any k\in N, p.v\int_{a}^{b}\frac{\cos kx}{t-x}dx=? We know that the Hilbert transform of cosine function is sine,see http://mathworld.wolfram.com/HilbertTransform.html. Now, we don't integral from -\infty \to \infty. We integral from a \to ... 1answer 254 views ### Integrating \sin(n\theta(x))/\sin(\theta(x)) for some function \theta(x) I have an indefinite integral of the form:$$ \int \frac{\sin(n\theta(x)))}{\sin(\theta(x))} dx. $$\theta is a function of x (and actually a complicated one). Is it possible to integrate it ... 1answer 118 views ### How to compute this distribution? My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that \Delta v is the distribution composed of the ... 1answer 49 views ### how to prove the convolution formular? let \overset{\backsim} {g}(x)=g(-x); suppose u,\phi,\psi always make the integral significant,E_n is the n-dimensional euclidean space. Then how to prove ... 1answer 160 views ### Bounds on integral I am calculating Fourier coefficients for certain functions and have come across an integral of the form$$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$where ... 1answer 5k views ### Criteria for swapping integration and summation order I have a function (a potential from an electrostatic potential via a Fourier series) in the form of$$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv ...
On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...