Differentiation under the integral sign
Let $\delta > 0$, $u \in L^1([-\delta,\delta])$ and a function $k: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies the following assumptions:$k \in L^1(\mathbb{R})$.$k$ is compactly supported on...
View ArticleRelation between roots of finite sum of exponentials and the finite sum of...
In my research, I am dealing with functions$$f(x) = \sum_{k=1}^N \frac{A_k}{B_k} \left(1- e^{-B_k x} \right), \quad g(x) = \sum^N_{k=1} \frac{A_k}{B_k}\left( \frac{1}{2}- \frac{1}{1 + e^{-B_k x}}...
View ArticleFix first and second derivatives at two points: is convexity possible?
Assume to have a smooth (or at least $C^2$) and convex real function $f$ such that:$f(x)>0$ for $x\in (a,b)$$f'(x)>0$ for $x\in (a,b)$$f''(x)>0$ for $x\in (a,b)$Now, fix an interval $[x_1,x_2]...
View ArticleConnectedness of a set formed by the complement of intersection of a bounded...
Let $W\subset \mathcal{R}^n$ be a bounded convex set. Let $y_0\in \partial \overline{W}$. I want to argue: there exists some $\delta>0$ such that $\forall 0<\delta'<\delta$,...
View ArticleIf $f(x)\sim g(x)$, is $f'(x)\sim g'(x)$ for $f,g\in C^{\infty}(\mathbb{C})$?
Consider a complex-valued function $f(\sigma+it)$ which satisfies $$f(\sigma+it)\sim g(\sigma+it),$$ as $|t|\to\infty.$ Must it be the case that $f'(\sigma+it)\sim g'(\sigma+it)$?Consider, for example,...
View ArticleIntersecting a convex set with a ball exactly at two points
In the following I am using the usual Euclidean norm. Let $W\subset\mathcal{R}^2$ be a non-empty bounded convex set. Let $y_0\in \partial W$ (boundary of $W$). I want to argue: there exists a...
View ArticleThe Fourier transform of $e^{-i/x}$
$\def\R{\mathbb R}$Question.Does anyone know what is the Fourier transform of$$ f(x)=e^{-i/x} $$on the real line? I would like to compute it explicitly, or to establish some properties to have a good...
View ArticleMinimum of set which has real numbers greater than 0
In a set:$S :=( x\in R: x > 0)$I know that infimum of this set is $0$. But what is the minimum of this set? My textbook defines the minimum as an element of the set. Thus it can’t be equal to $0$....
View ArticleIf I wanted to show that an isometry is always continuous, is this right?
So I need to show that an isometry is always continuous, (from $M$ to $N$ in this case) and my first thought was to show that for some $p,q\in M$, $\exists$ $\varepsilon\gt0$ such that $d_M(p,q)\lt...
View ArticlePolya's Condition for characteristic functions
Im reading the proof of the Pólya's condition ( Theorem 4.3.1) in the book "Characteristic Functions , by Eugene Lukacs 2ed".But i dont understand the following statament in the proof:Asume that:...
View ArticleConvolution with Dirac comb
I am trying to understand convolutions with distributions involving sums of delta functions. Consider the Dirac comb $\text{Sha} = \sum_{n \in \mathbb{Z}} \delta(x-n)$ and a function $f(x)$.The...
View ArticleIs topology just a generalization of real analysis?
While trying to learn undergraduate topology, I came across this lecture by Dr. Zimmerman who claims "Topology is a generalization of real analysis, a lot of topology anyway." They are obviously...
View ArticleShow measure of $\liminf$ is less than $\liminf$ of measure (Proof Verification)
If $(X,M,\mu)$ is a measure space and {$E_j$}, j=1 ,2....$\infty$.$\subset$ M, then $\mu$ (lim inf $E_j$) $\leq$ lim inf $\mu(E_j)$Proof:$\mu(\liminf E_j)=\mu(\left(\bigcup_{k=1}^{\infty}...
View ArticleA Counterexample in Fubini's Theorem by Kolmogorov
Let $E = [0,1]\times[0,1]$ and $f: E \rightarrow \mathbb{R}$ given by $f(x,y) = 2^n, \ \text{if}\ x \in [\frac{1}{2^n}, \frac{1}{2^{n-1}}] \ \text{and} \ y \in [\frac{1}{2^n}, \frac{1}{2^{n-1}}); \ \...
View Article$f$ is continuous if and only if $\text{osc}(f,x) = 0$ in a generic Hausdorff...
Let $(X,\tau)$ be a Hausdorff topological space and $f:X\rightarrow \mathbb{R}$ be a function. Let $x\in X$ and $U(x)$ being the of neighbourhood sets of $x$. Define the oscillation of $f$ at $x$ by...
View ArticleDoes zero derivative on a non-open connected set imply function is constant
The following is well-known:Theorem A: If $E$ is an open, connected subset of $\mathbb{R}^n$, then any function $f\colon E\to\mathbb{R}$ that is differentiable everywhere on $E$ with $f'=0$ must be...
View ArticleReflexivity of normed linear space ( or Banach space ) can be checked for...
Let $X$ be a normed linear space ( or Banach space ). The linear operator $J : X \to (X^{*})^{*}$ defined by$$ J(x)[\psi] = \psi(x) \ \operatorname{for all} x\in X , \psi \in X^{*}$$is called the...
View ArticleCould someone please give me some intution for this result for testing...
In class, we proved this following result for testing for uniform convergence of a sequence of functions without much elaboration:Theorem. A sequence of function $(f_n)$ is uniformly convergent iff:If...
View ArticleConvergence of a Specific Recursive Sequence
I'm trying to prove that the following recursive sequence converges to 1:$$x_1=0$$$$x_{n+1}=x_n-2\frac{e^{x_n}-ex_n}{e^{x_n}-e}$$The sequence has no fixed points, but...
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A question about interchanging order of supremum and limit
I encountered some gaps in Amann and Escher’s Analysis Ithe hypothesis of theorem3.2 is the function f is n-times continuously differentiable and its domain is a convex perfect subset of...
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