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Apr
2
accepted Counter-Example for Radon-Nikodym Theorem
Apr
2
asked Counter-Example for Radon-Nikodym Theorem
Mar
6
accepted Normalized partial sums of normal random variables are dense in $\mathbb{R}$
Mar
5
asked Normalized partial sums of normal random variables are dense in $\mathbb{R}$
Mar
2
awarded  Popular Question
Feb
28
accepted Defining weak* convergence of measures using compactly supported continuous functions
Feb
27
comment Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Yep, that is indeed true :)
Feb
27
comment Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Oops, you're right, it is only conditionally integrable and not absolutely integrable. My bad. There is an example though of a strictly non-negative function that I think is $L^{1}$ and doesn't vanish at infinity, it appears in Counterexamples in Analysis on page 45. Or at least it's an example of a continuous function with an improper integral on $[1,\infty)$ that doesn't vanish at $+\infty$.
Feb
27
comment Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Continuity enough isn't sufficient to ensure $f\in L^{1}$ vanishes at infinity, take for example $cos(x^2)$. Uniform continuity however as stated in the answer below does guarantee it.
Feb
27
comment Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Lovely! thanks so much :) I already knew that $L^{1}$ functions which are uniformly continuous must vanish at infinity, I didn't think about showing that integrable derivative implies uniform continuity.
Feb
27
accepted Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Feb
27
asked Bounding Fourier transform of $L^{1}$ functions that don't vanish at infinity.
Feb
26
comment Convergent sequence of characteristic functions with continuous integrable limit.
@Augustin how do I show that this $f$ defined in the statement is a density, I would have to show it's non-negative and integrates to 1 on the real line.
Feb
26
asked Convergent sequence of characteristic functions with continuous integrable limit.
Feb
24
comment Defining weak* convergence of measures using compactly supported continuous functions
@drhab I see that in page 60 it is defined for compactly supported continuous functions and at page 90 for bounded continuous function, there is no mention of it being equivalent. Obviously if $\mathcal{C}_{c}\left(\mathbb{R}\right)$ was dense in $\mathcal{C}_{b}\left(\mathbb{R}\right)$ (with respect to uniform norm) that would justify the equivalence of the definitions but I don't think that is true.
Feb
24
asked Defining weak* convergence of measures using compactly supported continuous functions
Feb
22
awarded  Nice Question
Feb
1
accepted Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
Feb
1
comment Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
@ByronSchmuland I'm not familiar with that result I'm afraid. Could you provide a reference please? Preferably one with a proof of the lemma. Thanks!
Feb
1
revised Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
added 14 characters in body; edited title