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Example of a periodic smooth function on $\mathbb{R}$ whose derivatives grow...

I'm surprised I can't find this anywhere, but can someone give me an example of a smooth (infinitely differentiable) periodic function on $\mathbb{R}$, but where the supremum of the derivatives is...

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Linear continuous map

Usually I know that in order to show that a map is linear I have to prove the additivity and homogenity. In most cases, I worked with functions that were given. In the next exercise, it is a bit...

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Is it possible to extend a uniformly continuous function $f$ with domain...

Let $\Omega\subset \mathbb{R}^n$ and $f:\Omega\rightarrow \mathbb{R}^m$ uniformly continuous. Prove that exists $\overline{f}:\overline{\Omega}\rightarrow \mathbb{R}^m$ uniformly continuous...

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y_n \text{ holds for arbitrarily large } n \Leftrightarrow \limsup\limits_{n\to\infty} \frac{x_n}{y_n} >1$" true?">Is "$x_n>y_n \text{ holds for arbitrarily large } n \Leftrightarrow...

Let $\{x_n\}_{n=1}^\infty \subseteq (0,\infty)$ and $\{y_n\}_{n=1}^\infty \subseteq (0,\infty)$ be some positive (possibly unbounded) sequences. I am trying to "cleanly" and "tightly" characterize the...

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Compute...

Compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+(x-y)^2+y^2)}dxdy$.I tried to do this by using polar coordinate. Let $x=r\cos t,\ y=r\sin t$, and...

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A $\sigma$-algebra $\mathcal{S}$ on $\mathbb{R}$ that is larger than...

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.The author wrote as follows on p.52 in this book:We have accomplished the major goal of this section, which was to show that...

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Nondecreasing interval on a locally absolutely continuous function $f: [a, b]...

For my research, I need to verify whether a function $f: [a, b] \to \mathbb{R}$ with $f(b) > f(a)$has a nondecreasing interval $(c, d) \subset [a, b]$ such that $f$ is increasing on that interval...

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A continuous function can be bounded from above by a smooth 'adapted' function?

Consider a continuous function $f:\mathbb{R}\to\mathbb{R}$. Is there a standard strategy that produces a function $g\in C^{\infty}(\mathbb{R},\mathbb{R})$, such that$f(t)\le g(t)$$\forall t$;$g(t)$ can...

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$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ then $f ' (0) =?$

Given$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ where $f$ and $g$ are continuous on $[0 , \infty ) , f >0 $ on $(0 , \infty)$ and $g>0 $ on $[0 , \infty )$ then $f'(0) $ is equal to$(a)\space \frac...

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Prove that a there is a measurable subset $ E_0 \subseteq [a, b] $ and...

I am working on my homework from Royden's Real Analysis:Let $E$ be a measurable subset of $[a, b]$, and define $ f: [a, b] \to \mathbb{R} $ by $ f = \chi_E $. For each $ \epsilon > 0 $, show that...

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How to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and...

How to calculate$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$$and$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$$by means of real methods?This question was suggested by Cornel the author of the book,...

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I'm just starting grad school and I'm already finding baby rudin's exercises...

I'm an economics bsc just starting grad school in quantitative finance, my first course is calculus, on week 1 we're studying the first two chapters of principles of mathematical analysis by Walter...

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Prove $\sum\limits_{i=1}^{N} m_i \ c_i \ \ln...

I am interested in solving a rather peculiar inequality that, like some of my previous posts, is physics-related because it comes from a physical condition that must be satisfied by a mathematical...

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Number of solutions for $N$-variable diophantine equation asserts this...

Let $a_1,a_2,\cdots,a_l$ be positive integers without a common factor different from $1$ and $A_n$ be the number of non-negative solutions of $\sum_{i=1}^la_ix_i=n$$\cdots\cdots$We suppose more: we...

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If the logarithm of two sequences gets close together, do the sequences get...

I should define what I mean by two sequences $(a_n)$ and $(b_n)$ that are close together. What I mean is that for any $\epsilon > 0$, $\exists N$ such that for $n \geq N$ we have $|a_n-b_n| <...

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On the Constant Rank Theorem and the Frobenius Theorem for differential...

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the...

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Convolution of non negative, symmetric, decreasing functions

Suppose $f,g\in L^1(\mathbb{R})$ are non negative, even and decreasing on $[0,\infty)$. Prove that their convolution $f * g$ is even and decreasing on $[0,\infty)$. Attempt: It is easy to show that $f...

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Analyzing the asymptotic behavior of a given function $f(\mu, t)$ as $t \to...

Consider the following improper integral$$f(\mu, t) = \frac{2}{\pi} \int_0^\infty \frac{\sin u}{u} \frac{\left(U+u\right) \exp\left(-\frac{\mu}{t} \, u\right) - 2U \exp\left(-\frac{\mu}{t} \, U...

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Kolmogorov–Arnold representation for permutation invariant functions

Kolmogorov Arnold reperesentation theorem states that (the extended version by Lorentz) every multivariate continuous function $f:[0,1]^n\rightarrow \mathbb{R}$ can be written in the folowing form,...

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Given two finite signed measures that are equal on h-intervals. Are they equal?

I come up with this question when I was trying to understand Folland's proof of Theorem 3.29.Theorem: if $F\in NBV$, there is a unique complex Borel measure $\mu=\mu_r+i\mu_i$ such that...

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