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...
View ArticleLinear 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...
View ArticleIs 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...
View Articley_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...
View ArticleCompute...
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...
View ArticleA $\sigma$-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...
View ArticleNondecreasing 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...
View ArticleA 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)$$\forall t$;$g(t)$ can...
View Article$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ then $f ' (0) =?$
Given$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ where $f$ and $g$ are continuous on $[0 , \infty ) , f >0 $ on $(0 , \infty)$ and $g>0 $ on $[0 , \infty )$ then $f'(0) $ is equal to$(a)\space \frac...
View ArticleProve that a there is a measurable subset $ E_0 \subseteq [a, b] $ and...
I am working on my homework from Royden's Real Analysis:Let $E$ be a measurable subset of $[a, b]$, and define $ f: [a, b] \to \mathbb{R} $ by $ f = \chi_E $. For each $ \epsilon > 0 $, show that...
View ArticleHow to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and...
How to calculate$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$$and$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$$by means of real methods?This question was suggested by Cornel the author of the book,...
View ArticleI'm just starting grad school and I'm already finding baby rudin's exercises...
I'm an economics bsc just starting grad school in quantitative finance, my first course is calculus, on week 1 we're studying the first two chapters of principles of mathematical analysis by Walter...
View ArticleProve $\sum\limits_{i=1}^{N} m_i \ c_i \ \ln...
I am interested in solving a rather peculiar inequality that, like some of my previous posts, is physics-related because it comes from a physical condition that must be satisfied by a mathematical...
View ArticleNumber of solutions for $N$-variable diophantine equation asserts this...
Let $a_1,a_2,\cdots,a_l$ be positive integers without a common factor different from $1$ and $A_n$ be the number of non-negative solutions of $\sum_{i=1}^la_ix_i=n$$\cdots\cdots$We suppose more: we...
View ArticleIf 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| <...
View ArticleOn 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...
View ArticleConvolution 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...
View ArticleAnalyzing 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...
View ArticleKolmogorov–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,...
View ArticleGiven 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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