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).

7,480 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
65
votes
0answers
1k views

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

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 ...
22
votes
0answers
564 views

What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which ...
22
votes
0answers
914 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: (i)...
21
votes
0answers
534 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
20
votes
1answer
545 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
17
votes
0answers
287 views

Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
17
votes
0answers
1k views

Errata for Dieudonné's Treatise on Analysis volume 2 second edition

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
14
votes
0answers
656 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
11
votes
0answers
256 views

Tricky surface integral of vector field

We have the embedded surface $S= \{(x,y,z)\in \mathbb{R}: z = e^{1-(x^2 + y^2)^2}, z>1\}$ and the vector field $\mathbf F:\mathbb{R}^3\to \mathbb{R}^3; (x,y,z)\mapsto (x e^{y^2}, 2ye^{x^2}, 5-3z) $....
11
votes
0answers
572 views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
11
votes
0answers
643 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
11
votes
1answer
969 views

Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
11
votes
0answers
1k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
10
votes
1answer
226 views

A problem about periodic functions

Suppose that $f(x):\mathbb{R}\rightarrow \mathbb{R}$ is a periodic function with a minimal positive period $T$. Can $g(x)=f(x^2)$ be periodic? I know it is impossible if the condition $\forall x \in (...
10
votes
0answers
637 views

Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
10
votes
0answers
833 views

Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
10
votes
0answers
817 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2\left(\frac{\omega_{12}}{2}\right)\sin^2\left(\frac{\omega_{23}}{2}\right)d\omega_{23}d\omega_{12}$$ where: ...
9
votes
0answers
137 views

What exactly does curl measure? What is rotating about what?

This question is a bit long, the next two paragraphs give some context, but you can skip it. Thank you. I have seen many different explanations for the meaning of curl, or what exactly does it ...
9
votes
0answers
1k views

What is a fewnomial?

I came across the theory of "fewnomials" (by Khovanskii), which (I guess) are related to polynomials. However, I was surprised that there is no single question on stackexchange concerning fewnomials, ...
9
votes
0answers
476 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
9
votes
0answers
770 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ can ...
9
votes
1answer
213 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see http://en....
9
votes
0answers
254 views

$W^{1,p} $ and $W^{2,p}$ Estimates.

In the beginning of section 4 in here the author says that one can easily adapt the methods in the preceding section to obtain $W^{1,p}$ estimate. I'm trying to do this. I think the following: the ...
9
votes
0answers
248 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
9
votes
0answers
370 views

How can we prove a simple case of the High Indices Theorem?

Let $(a_n)$ be a sequence of real numbers such that $$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$ converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
8
votes
0answers
120 views

How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $f$ be the function defined on all of $\mathbb{R}$ by the formula $$ f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right). $$ How to show (rigorously but through elementary ...
8
votes
0answers
113 views

An attractor for blow-up solutions to a cubic oscillator

(Related to this MathOverflow question). Consider the nonlinear ODE $$\tag{1} \frac{d^2u}{dt^2}+u=u^3, \qquad t\in\mathbb R,$$ which has the conserved quantity $$\tag{2} E=\frac12 u'^2+\frac12 u^2 -...
8
votes
0answers
151 views

Defining an unusual subspace of $c_0$

This is going to be a long post, so I'm giving a description first: I recently came across the following exercise: Let $(X,\mathcal{A},\mu)$ be a measure space. If $(f_n)\subset L^p(\mu)$ and $f\in L^...
8
votes
0answers
243 views

Generalizations of Integrals

I've been studying dimensional regularization recently and found that some people view the divergent integrals, which are replaced by certain finite expressions in the regularization procedure, as ...
8
votes
0answers
483 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
8
votes
1answer
962 views

Exercise 6.9 in Rudin's RCA (Real and Complex Analysis)

The following is an exercise 6.9 in Rudin's Real and Complex Analysis: Suppose that $\{ g_n \}$ is a sequence of positive continuous functions on $I=[0,1]$, that $\mu$ is a positive Borel measure on $...
8
votes
0answers
404 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n \...
8
votes
0answers
257 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
8
votes
1answer
859 views

Lebesgue point of density on $[0,1]$ and Dynkin's theorem

The problem defines a density point $x\in[0,1]$ for a Borel set $A\subset [0,1]$ if $$ \lim_{\varepsilon \rightarrow 0^+} \frac{\mu([x-\varepsilon,x+\varepsilon]\cap A)}{2\varepsilon}=1.$$Denote all ...
7
votes
0answers
134 views

Are there any other fields other than $\mathbb{R},\mathbb{C}$, rich enough to have analysis built on them?

I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties: Algebraic structure: ...
7
votes
0answers
284 views

Show the sequence $\{{\sqrt{5}}~,{\sqrt{5+{\sqrt5}}}~,\sqrt{5+\sqrt{5+\sqrt{5}}}~,…\}$ converges and find its limit.

Show the sequence $\bigg\{{\sqrt{5}}~,{\sqrt{5+{\sqrt5}}}~,\sqrt{5+\sqrt{5+\sqrt{5}}}~,...\bigg\}$ converges and find its limit. Attempt : Let $a_{1}=\sqrt{5}$ and $a_{n+1}={\sqrt{5+a_{n}}}$ for $n=...
7
votes
0answers
647 views

Are there any simple examples of Kolmogorov-Arnold representation?

I had never heard of the Kolmogorov-Arnold Representation Theorem before. It states roughly that any multivariable function can be represented by repeatedly adding a single variable function whose ...
7
votes
0answers
278 views

A sufficient condition for a sequence to converge if arithmetic mean of the sequence converges?

We have a well-known conclusion: If a sequence $\{a_n\}_{n\in\mathbb{N}}$ converges, then the arithmetic mean $\frac{S_n}{n}$ (where $S_n=\sum\limits_{k=1}^na_k$ is the nth partial sum) converges to ...
7
votes
0answers
1k views

If $E$ is a totally bounded subset of a metric space $X$. Then, any subset of $E$ is totally bounded.

Proof: Let $D \subset E$, where $E \subset X$ and ($X,d$) is a metric space. Suppose that $E$ is totally bounded. That is: for all $\varepsilon > 0$, there exist finitely many points $x_1, \ldots , ...
7
votes
0answers
327 views

Improper Integral $\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$

This integral is from integral Find $$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$$ I have get $$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\...
7
votes
1answer
72 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...
7
votes
0answers
184 views

prove uniqueness from orthogonality relation

The problem: we have two functions $f(x), g(x)\in C^{1}[-\pi, \pi]$, and we know that \begin{align} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \cos\left(ky\right) \sin\left(k\left\lvert y-z\right\rvert\...
7
votes
0answers
274 views

Prove: $\frac{p}{2\pi}\int_{-\infty}^{+\infty}\frac{\sin xt}{t\cdot \sin\frac12pt}\sin([\frac xp]+\frac12) pt \, \mathrm dt=\cdots$

Suppose $p>0$, define that $$ g(x)=\begin{cases} p\left\lfloor\frac xp\right\rfloor+\frac p2,x\geqslant0\\\\-g(-x), x<0\end{cases}$$ Prove for all $x$, $$ \frac{p}{2\pi}\int_{-\infty}^{+\...
7
votes
0answers
146 views

Approximating intervals and squares by increasingly dense disjoint finite sets with special properties

Apologies for the length of the question. Consider interval $I=[0,1]$. For any $n \in \mathbb{N}$ we can always find two finite sets $S_{1n} \subset I$ and $S_{2n} \subset I$ such that: a) $S_{1n}\...
7
votes
1answer
212 views

Another conditionl leading to irrationality of $\sum _{k=1}^ \infty \dfrac 1{n_k}$?

If $\{n_k\}$ is a strictly increasing sequence of positive integers such that $\lim \inf _{k \to \infty} n_k ^{1/2^k} >1$ and $\lim _{k \to \infty} n_k^{1/2^k}$ does not exist , then is it true ...
7
votes
0answers
1k views

Rudin's Rank theorem

Rudin states the following: 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ is a $C^1$ mapping of an open set $E\subset R^n$ into $R^m$, and $F'(x)$ has rank $r$ for ...
7
votes
0answers
613 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
7
votes
0answers
913 views

Prob. 10, Sec. 3.2 in Erwin Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwin Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex inner ...
7
votes
0answers
827 views

Conditions on $f$ such that separate continuity implies joint continuity

Consider a function $f : X\times Y \to Z$ where all spaces are compact metric spaces. Assume further that $f_x: y \mapsto f(x,y)$ and $f_y: x \mapsto f(x,y)$ are continuous. I am looking for ...
7
votes
0answers
1k views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...

1
2 3 4 5
150