Questions tagged [uniform-convergence]
For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].
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Uniform convergence on compact subsets implies weak convergence
Let $f_n$ be a sequence in $L^p(\mathbb{R})$ with $1 < p < \infty$ so that $f_n$ converges uniformly to $f$ on every compact subset of the real line. Find whether or not $f_n$ converges weakly ...
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Does uniform convergence to zero function on every closed and bounded interval imply uniform convergence on $\mathbb{R}?$ [duplicate]
I am stuck to the following question while self studying Real Analysis.
Let $f_n: \mathbb{R} \rightarrow \mathbb{R},$ and suppose that $f_n \rightrightarrows 0 \ (f_n$ converges uniformly to the zero ...
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Find an interval for which $f_n(x) = \frac{x}{1+nx}$ does not converge uniformly
I am studying for my Real Analysis course and one of my practice problems asks us to "prove the sequence of functions $f_n(x) = \frac{x}{1+nx} \to f$ uniformly on certain intervals."
I've ...
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Does Convergence of Arc Length Imply Uniform Convergence?
Define $L_f(a,b)$ denote the arc length of the graph of the function $f$ on $(a,b)$.
For a sequence of functions $f_n(x):D\to \mathbb{R}$ that converges to $f(x)$, even if $f_n$ converge uniformly to $...
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What $a,b$ such that $\sum\limits_{n\ge 1}\frac{x^b}{n(1+n|x|^a)}$ converge uniformly on $\mathbb{R}$.
I was solving this problem
Prove that $\displaystyle\sum\limits_{n\ge 1}\frac{x}{n(1+n x^2)}$ converge uniformly on $\mathbb{R}$
I was able to prove this mainly by using Am-Gm inequality: $1+nx^2>...
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Swapping sum and integral with an infinite series of modified Bessel functions
I am studying the following integral
\begin{align}
\int_0^{\infty} I_1(\sqrt{x}) \sum_{k=1}^{\infty} \Big( k K_1(k\sqrt{x}) - k^2 \sqrt{x}\, K_0(k\sqrt{x})\Big ) dx
\end{align}
where $I_1$ and $K_\nu$ ...
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How do you think about uniform continuity?
My background is in physics, so I never had a proper course in either real or complex analysis; topics like uniform convergence weren't touched upon. I really like analysis though, so for the last few ...
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Does this function disprove the statement that uniform convergences preserves no jump discontinuities?
This is question 5 from chapter 4 from Pugh's Real Mathematical Analysis. Part (f) asks if uniform convergence preserves "no jump discontinuities." I believe that I have created a function ...
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Differentiability of the series $\sum_{n=1}^\infty \frac{\sin(n^2x)}{n^2}$.
I was playing around in Desmos and found the following interesting series similar to the Weierstrass function: $$\sum_{n=1}^\infty \frac{\sin(n^2x)}{n^2}$$
I noted that this is uniformly convergent (...
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Uniform convergence in the central limit theorem
I was reading the following notes, https://www.cs.toronto.edu/~yuvalf/CLT.pdf, on the central limit theorem. I am a little confused about what the author says on page two,
"The exact form of ...
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Question in the proof of Hankel's integral representation of the Bessel function of the first kind
I am trying to understand the proof of Hankel's integral representation of $J_\alpha(x)$:
$$ J_\alpha(x) = \frac{(x/2)^\alpha}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-\alpha -1} \exp\left(t-\frac{x^2}...
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Why count both horizontal and vertical parts in the staircase paradox?
I have a question regarding the description of the staircase paradox on Wikipedia.
As I understand it, the horizontal segments form a "staircase" that approaches the diagonal.
Considering ...
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Is this proof that $\sum_{n=0}^\infty\frac{\sin2nx}{(2+\sin x)^{n^2}}$ does not converge uniformly on intervals containing $-\frac{\pi}{2}$ correct?
I have provided an answer to this question prooving that $\displaystyle f(x)=\sum_{n=0}^\infty\frac{\sin2nx}{(2+\sin x)^{n^2}}$ does not converge uniformly on any interval containing $\displaystyle-\...
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Show $\displaystyle\sum_{n=0}^\infty\frac{\sin(2nx)}{(2+\sin x)^{n^2}}$ does not converge uniformly on every interval containing $x=-\pi/2$
I want to show that the series $$\sum_{n=0}^{\infty}\frac{\sin(2nx)}{(2+\sin x)^{n^2}}$$
does not converge uniformly on every interval containing $x = -\frac{\pi}{2}$.
I managed to prove uniform ...
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Uniform convergence of the integral of a product if only the integrals of the factors converge? [closed]
I am trying to establish the following result:
Let $\mu_n$, $n\in \mathbb{N}$, and $\mu$ be probability measures such that for the associated measure-generating functions $F_n, F$ it holds that $F_n$ ...
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