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I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it having 6 upvotes and two favourites] received no answer even after a 300-point bounty. Therefore, I am re-posting it here. Let $ f \colon \mathbb{R} \to \mathbb{R}$ be a bounded $ L^1$ function that is piecewise-smooth (with the boundaries of the piecesRead more

Let $ H_{d,g,3}$ denote the open subscheme of non-singular irreducible curves in $ \text{Hilb}^{dn+1-g}_{\mathbf{P}^3}$ . Mumford showed that when $ g=24$ and $ d= 14$ , the open subscheme has an irreducible component that’s generically non-reduced. Are there other examples for lower genera? In particular, what is the smallest genus for which one can findRead more

Problem. Does every compact metric space of finite packing dimension admit a bi-Lipschitz embedding into the Hilbert space $ \ell_2$ ? The packing dimension of a compact metric space $ (X,d)$ in the (finite or infinite) number $ $ Dim(X)=\limsup_{\varepsilon\to 0}\frac{\ln N_\varepsilon(X)}{\ln(1/\varepsilon)},$ $ where $ N_\varepsilon(X)$ is the cardinality of the smallest cover of $Read more

In a lecture on Quantum mechanics, the professor concluded that if $ a$ is a linear operator with $ [a, a^\dagger] = 1$ , where $ a^\dagger$ is the adjoint of $ a$ and $ [a, a^\dagger] = aa^\dagger – a^\dagger a$ , and if $ a^\dagger a$ has an eigenvalue, then vectors of formRead more

Background: Let $ D$ be a Klein Four group and consider free product $ Z/2Z\star D=<a,b,c,d|a^{2}=b^{2}=c^{2}=d^{2}=bcd=1>$ . Now we consider group algebra generated by $ Z/2Z\star D$ with inner product $ <f,g>=tr(g^{*}f)$ . After taking completion, we have a Hilbert space. Question: Now Let $ u=\frac{b+c+d-1}{2}$ , consider two closed subspaces of the above HilbertRead more

Let $ (\Omega,\mathcal A,\operatorname P)$ be a complete probability space $ (\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $ (\Omega,\mathcal A)$ $ U$ be an infinite-dimensional separable $ \mathbb R$ -Hilbert space $ Q$ be a nonnegative and self-adjoint nuclear operator on $ U$ $ U_0:=Q^{1/2}U$ be equipped with $ $ \langle u_0,v_0\rangle_{U_0}:=\langleRead more

Prove that and isometry $ \varphi$ in a finite-dimensional real Hilbert space with $ \varphi(0)=0$ is linear. Im stuck whit this exercise. My work so far: Let $ H$ our Hilbert space. Because $ \varphi(0)=0$ and it is an isometry we have that $ $ \|\varphi(x)-\varphi(0)\|=\|x-0\|\implies\|\varphi(x)\|=\|x\|,\quad \forall x\in H\tag1 $ $ Thus $ $ \|\varphi(x)-\varphi(y)\|=\|x-y\|=\|\varphi(x-y)\|,\quad\forallRead more

As part of the existence of the Hilbert class field, we have the following equivalence : Let $ k$ be a number field. Then the class group of $ k$ is non trivial if and only if $ k$ has an abelian extension which is unramified everywhere. How much of that needs advanced Class FieldRead more