Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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A question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis

I have a question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis (3rd edition, page 56). Here Rudin explicitly requires that $|f| \leq 1$. But I can not see why this requirement is necessary. I can not even see why $|f|$ should…
user55329
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a function in the unit ball of C(X) where X is a compact space is a limit of convex combinations of extreme points

Suppose $f(x)$ is a function in $C(X)$ such that $\|f\|<1-\frac{2}{n}$. Then there exist n extreme points of the unit ball of $C(X)$, $g_1,\ldots,g_n$ such that $f(x)=\frac{1}{n}(g_1(x)+\cdots+g_n(x))$. I know that $g$ is an extreme point of the…
john
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Why can't $\theta(0)$ be $0$? (tempered distributions)

I'm reading about the Littlewood-Paley decomposition, but there is a definition I can't understand, it says: We denote by $S'h(\mathbb{R}^d)$ the space of tempered distributions $u$ such that $\displaystyle\lim_{\lambda\to\infty}\|\theta(\lambda…
user38448-po
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Is there a closed form expression for the following series?

Is there a closed form expression for $$ \sum_{1 \le k \le n}ke^{-t}\left(e^{k+\frac{1}{2k^3}}-e^{k-\frac{1}{2k^3}} \right)? $$ this series emerge in computation output of this system via convolution of input $u(t)$ and impulse response (inverse…
mary
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Prove that $\{f * \phi _n\}$ uniformly converges to $f$.

Let $\phi _n : \mathbb{R} \to \mathbb{R}$ be a sequence of functions with the following properties: 1) For all $n \in \mathbb{N}$ we have $\phi _n (x) \geq 0$ for all $x \in \mathbb{R}$ and $\phi_n (x) = 0$ for all $|x| \geq 1$. 2) $\int_{-…
Compact
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For which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1])$?

I want to find out for which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1]), p \in \mathbb{R}_{\geq 1} \cup \{\infty\}$. I first let $p,q \geq 1$ and I looked at the case where $q > p$. I defined $\tilde{f} := |f|^p$ and…
Huy
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Is there a Geometrical Interpretation for Pseudo-Monotonicity?

Suppose $X$ is a real Reflexive Banach space and $f:X\rightarrow X^\star$ a pseudo-monotone map (see here for a definition of pseudo-monotone). Is there a geoemetric interpretation for this definition. For example, we know that a differentiable $f:…
Tomás
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analemma curvature

By making some account of astronomy I came across the parametric curve of the analemma: $$ x(t) = \arcsin(s \sin(t)), y(t)= \arctan\left[\frac{(1-c)\tan(t)}{1+c \tan(t)^2}\right]$$ where $ s = \ sin (23,5 °) $, $ c = \ cos (23,5 °) $ (the sine and…
user273366
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What is the proper convergence proof for this sequence?

Limit of $n^{1/2}(n^{1/n}-1)=0$, as $n$ approaches infinity. I need a strict math proof that this sequence converges to zero, but without using L'Hôpital's rule, because I am not allowed to use it yet. Thanks in advance.
Nikola Zoranovic
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How can I show the Coercivity of this function?

Let $S$ be the set of real positive matrices, $\lambda>0$ and $f:S\rightarrow\mathbb{R}$ defined by $$f(X)=\langle X,X\rangle-\lambda\log\det(X) $$ where $\langle X,X\rangle=\operatorname{trace}(X^\top X)$. How can one show that $f$ is coercive?
Tomás
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Fourier transform of $ |x|^{-\alpha} $

How to get the Fourier tramsform of $ |x|^{-\alpha} $ ; $ |x| = \sqrt{x_{1}^{2}+ ...+ x_{n}^{2}}$ ($x $ is multivariable). I really stuck here, I want to derive the Riesz potential using Fourier transform, I found in Wikipedia that $$…
abdullah azzam
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The Boundary of derivative

$y=f(x)$ is a $C^{\infty}$ function defined in $\mathbb{R}$,for any $k\in \mathbb{Z}^+$,we let $M_k=\mathop{\sup}_{x\in\mathbb{R}}|f^{k}(x)|$. $m,n \in \mathbb{Z}$ and $0\le m
Ryze
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Does the sum of $\frac{\pm a}{\log(1+ax)}$ have at most finitely many zeros?

Suppose $a_1, a_2, ..., a_k\in\mathbb{R}$. Is it true that $f(x)=\sum\limits_{1 \leq i \leq k}{\dfrac{\pm a_i}{\log(1+a_ix)}}$ has at most finitely many zeros (on the domain where $1+a_ix>0$ for all $i$) for every sign combination? I have plotted…
Strin
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How to get a bound of this function?

let $$ f(x)=\prod_{i=0}^n\left(x-\frac{i}{n}\right) $$ where $x\in[0,1]$. How to deduce a bound for it? Particularly, I want to prove $$ |f(x)|\leq\frac{(n+1)!}{n^{n+1}} $$
hxhxhx88
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Two-point boundary value problem: $u'' = -\lambda \cos u$

Suppose that we have the following $$ u'' = -\lambda \cos u $$ with $u(0) = u(1) = 0$. How can it be shown that if $|\lambda|$ is sufficiently small then the problem above has a unique solution? The hint is to reformulate the problem as a nonlinear…
Stan
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