5 votes
Accepted

Integral means vs infinite convex combinations

No. Let $(X, \cal A, \mu)$ be $[0,1]$ with Lebesgue measure. Let $E = L^2[0,1]$ with inner product $\langle \alpha,\beta\rangle := \int \alpha(t)\overline{\beta(t)}\;dt$. Define $f : [0,1] \to L^2[0,1]...
Gerald Edgar's user avatar
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4 votes

Integral means vs infinite convex combinations

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $...
an_ordinary_mathematician's user avatar
3 votes
Accepted

Concentration of measure on spheres with respect to a unitary of trace approximately zero

This is indeed true. Assume WLOG that the matrices are diagonal. A useful way to handle the uniform measure on the sphere is that if you let $X_1,\ldots,X_n$ be iid (complex) Gaussians, then the ...
Marcus M's user avatar
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