Is this a good construction of real exponentiation?...
In a previous post I tried to construct real exponentiation, i.e. the traditional "$b$ to the power of $n$" $b^n$ operation. I stupidly forgot that $b^\frac{1}{n}=\sqrt[n]{b}$ and is undefined for...
View Article$ f:X\to Y $ is continuous on $ X $ iff $ \forall A\subset X $, $...
I have always thought this result as a generalised version of the sequential criterion of continuity. Surely, $ \bar{A} $ contains all the convergent sequences from $ A $, and the result says that the...
View ArticleSuper-level sets of Hardy–Littlewood maximal function are open?
I am working on the book Measure, Integration and Real Analysis by Sheldon Axler. I am stuck on Problem 9 of Section 4A. For $h: \mathbb{R} \to \mathbb{R}$ Lebesgue measurable, $h^*$ is defined as...
View ArticleShow Fejer kernel on the real line is good, without using trignometric...
This is from page 163 of Stein's Fourier Analysis. Fejer kernel on the real line is defined by$$ \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2$$When $t=0$, $\mathcal{F}_R(t)=R$.I...
View ArticleSolve for $m(t)$ in the integral transform $\int_0^1(1-t^n) m(t)...
Background(You can skip this part, but maybe you find it interesting.)Is $ \displaystyle f_1(x,v) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^v } > 0 $ for all real $x$ and $0<v<1$ ?Lets start...
View ArticleApplying Banach contraction principle to square integrable functions on...
I am reading "Application number 3" of Banach contraction theorem from this article. I want to replace $L(a,b)$ in this example in the link with $L(0,\infty)$ so that. So we need to apply Banach...
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Geometric interpretation of the heat ball
I'm reading the section on the Heat equation in Evans' book and there is something I don't quite understand.He defines the heat ball as$$E(x, t, r) = \left\{(y, s) \in \mathbb R^n \times \mathbb...
View Articledivergence of sum ∑a_i+∑1/b_i=∞ based on two sequences ⇒ divergencence of...
A given series $(x_i)_{i=1}^{\infty}$, is split into $(a_i)$ and $(b_i)$ with$$ (x_i) = (a_i) \cup (b_i) ,$$ such that$$\lim a_i = \infty\text{,}$$$$\lim b_i = 0.$$If then $\sum_{i=1}^{\infty} a_i +...
View ArticleIs there an example of a TVS on which there is no nonzero continuous linear...
Let $f: X \rightarrow K$ be a nonzero continuous linear functional on a topological vector space (TVS).Prove that the set $U = \{x \in X: f(x) < 1\}$ is an open convex set in $X$ containing 0.Bonus:...
View ArticleEllipse: $x$ corresponding to minimum distance between $(x,y)$ and one focus
Consider an ellipse with foci in $(-c, 0)$ and $(c, 0)$, where $a$ is the length of the semimajor axis. Consider a point $(x,y)$ belonging to the ellipse. The point verifies the equation:$$\sqrt{(x -...
View ArticleWhen can the Weierstrass transform be represented as $e^{D^2}$?
The Weierstrass transform$W$ of $f:\mathbb{R}\to\mathbb{R}$ can be defined as the convolution of $f$ with a Gaussian:$$W[f](x) = \frac1{\sqrt{4\pi} } \int_{\mathbb{R}} dy\, f(x-y) e^{-y^2/4}.$$It's not...
View ArticleAnalytic sets have perfect set property (Kechris)
As title says. I’m trying to learn some descriptive set theory but I don’t quite see this. I want to use the following:Given $X, Y$ Polish spaces, $f:X\to Y$ continuous, if $f(X)$ is uncountable there...
View ArticleDecay Rate Fourier transform: $L^2$ function that is odd and compactly supported
Consider $f \in L^1(\mathbb R)$ and it's Fourier transform $\hat f : \mathbb R \rightarrow \mathbb C$ defined by$$\hat f(\xi) = \int_{-\infty}^{+\infty} e^{-ix\xi}f(x)dx.$$It is known (Riemann-Lebesgue...
View ArticleAn improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$
How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$.I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based on...
View ArticleAgain on the concavity of the logarithm
I'm asking a question again on the concavity of the logarithm, but this is not a duplicate. Rather, an attempt to give a proof, or at least that was the idea.In trying, I hit a wall of question marks,...
View Articlecomputing $\frac{1}{2\pi...
I was reading this and got stuck right before Remark $1$.Let$$F(t)=\frac{1}{2\pi i}\int\limits_{(1)}\frac{e^{(1-t)s}}{s}\Psi\left(\frac{\Im (s)}{\log x}\right)ds$$where $\Psi$ is a smooth cut off...
View ArticleExistance of tubular neighborhood
Let $S\subset \mathbb{R}^n$ be a compact oriented hypersurface of class $C^k$, $k\geq 2$. There is a neighborhood $V$ of $S$ in $\mathbb{R}^n$ and a number $\varepsilon >0$ such that the map $F(x,t)...
View ArticleHow to construct real exponentiation?
I have been trying to rigorously define real exponentiation. Online there doesn't seem to be ANY definition of real exponentiation that covers every case of base and exponent.In school and on...
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Does the limit $\lim_{x\to1^{-}}(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n}$ exist?
Let the lacunary power series $$f(x):=(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n},0<x<1.$$ We use Matlab to draw the function $f(x)$ graph, which shows that the function $f(x)$ has limits when...
View ArticleAnalysis book for someone with no experience with mathematical proofs? [closed]
I have no experience with mathematical proofs. I would like advice on which books I could use to start my own studies in real analysis which would also teach me from scratch about mathematical proofs...
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