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).

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18
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371 views

Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...
18
votes
0answers
379 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 ...
15
votes
0answers
294 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 ...
14
votes
0answers
543 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: (...
14
votes
0answers
798 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 ...
12
votes
0answers
245 views

Finding an example of nonhomeomorphic closed connected sets

Question: Find two closed, connected subsets in $\mathbb{R}^2$, $A$ and $B$, such that $A$ is not homeomorphic to $B$, but there is a continuous bijection $f:A \rightarrow B$ and a continuous ...
12
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0answers
129 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 $x$, $$\displaystyle\frac{p}{2\pi}\int_{-\...
12
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0answers
219 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
12
votes
0answers
1k views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
11
votes
0answers
424 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
0answers
314 views

Norm Inequality on a Compact Riemannian Manifold

Consider the following problem: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality $$\Delta u \geq -...
10
votes
0answers
926 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 ...
9
votes
0answers
286 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a $k$-...
9
votes
0answers
192 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
499 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 ...
9
votes
0answers
231 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 ...
8
votes
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143 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\...
8
votes
0answers
212 views

Brezis Exercise 3.27 Extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
7
votes
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128 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
0answers
222 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 ...
7
votes
0answers
361 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 ...
7
votes
0answers
290 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 $x_{0}$ ...
7
votes
0answers
141 views

Product of the first n cyclotomic polynomials.

Let $$F_n(\alpha) = \prod_{k = 1}^n \Phi_k(e(\alpha))$$ where $e(\alpha) = e^{2\pi i\alpha}$ It is clear that $F_n(\alpha) = 0$ iff $\alpha = \frac{a}{q}$ for relatively prime $a, q$ s.t. $q \le n$. ...
7
votes
0answers
208 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...
7
votes
0answers
188 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 \...
7
votes
0answers
193 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 ...
7
votes
0answers
295 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: $$\int_{0}^{2\pi}\frac{1}{\sqrt{a^2-b^2\cos^{2}2\phi}}\exp{\left(-\frac{(x-c\cos\phi)^2}{a+b\cos2\phi}-\frac{(y-d\sin\phi)^2}{a-b\cos2\phi}\...
7
votes
0answers
251 views

Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$

I am trying to solve a qualifying exam problem. I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup E_k\...
7
votes
0answers
302 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 ...
6
votes
0answers
134 views

Me vs. Wikipedia (Lacunary function)?

I was recently reading this wikipedia page: https://en.wikipedia.org/wiki/Lacunary_function and found atleast the example they are giving must be wrong because I have kind of managed to analytically ...
6
votes
0answers
146 views

Proving the function $ f $ is continuous on $ [0,1] $

I'm trying to prove that the following function $ f $ is continuous on $ [0,1] $. The function $ f:[0,1]\rightarrow [0,1] $ is defined as follows. Let $ x\in [0,1] $. Then $ x= \sum\limits_{n = 1}^\...
6
votes
0answers
152 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ~~~r>0,-\frac{\pi}{2}<\theta<\frac{\...
6
votes
0answers
276 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
6
votes
0answers
149 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
6
votes
0answers
155 views

Prove $\ell_1$ is complete

I try to prove that $\ell_1$, the space of absolutely convergent sequences in $\mathbb{C}$ with norm $\| x \| = \sum_{k=1}^{\infty} |x_k|$, is complete. I already proved that, if $\{ x_n \}$ is a ...
6
votes
0answers
768 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...
6
votes
0answers
104 views

Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
6
votes
0answers
390 views

Are these sets in $\mathbb{R}$ open and/or closed: $\{\frac{1}{n} : n \in \mathbb{N}\}$, $\{0\}\cup \{\frac{1}{n} : n \in \mathbb{N}\}$ and $[0,1)$.

In $\mathbb{R}$, are these sets open? Are they closed? $A = \{\frac{1}{n} : n \in \mathbb{N}\}$ $B = A \cup \{0\} $ $[0, 1)$ My thoughts: $A$ is not open as if we have an open ball with $r > ...
6
votes
0answers
234 views

Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$

Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$ For all $ n\in \mathbb{N} $ let $$ F_n = \{ f: \exists ...
6
votes
0answers
671 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f{} = 2\pi+2\tan^{-1}(y,x)$ $y = -...
5
votes
0answers
48 views

Question regarding continuity ($\epsilon$-$\delta$)

In this post: Proof continuity of a function with epsilon-delta people explained how to proof that $$f(x) = \frac{x-1}{x^2+1}$$ is continuous in $x_0 = -1$. However, my solution is the following: $$|f(...
5
votes
0answers
56 views

Schwartz functions dense?

I want to show that the Schwartz functions are dense in $$\left\{f \in L^2; \int |x|^2 \left|f(x)\right|^2 dx + \int |\xi|^2 \left|\hat{f}(\xi)\right|^2 d \xi < \infty\right\}$$ where the norm ...
5
votes
0answers
60 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
5
votes
0answers
82 views

Generalization of FTA

I'm sure that this is not any hypothesis, but following came to my mind when I was reading complex analysis. Consider a function $f(z)=z^n+g(z)$, where $g(z)$ is continuous (not necessary holomorphic)...
5
votes
0answers
65 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, z\in\...
5
votes
0answers
66 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
5
votes
0answers
68 views

On the form of injective function $f:\mathbb R^+ \to \mathbb R^+$ such that $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ is a norm

Let $f:\mathbb R^+ \to \mathbb R^+$ be injective such that on $\mathbb R^n , n>1)$ , $\|u\|:=f^{-1}\Big(\sum_{i=1}^nf(|u_i|)\Big)$ , where $u=(u_1,u_,\ldots,u_n)$ , defines a norm ; then is it true ...
5
votes
0answers
66 views

Efficient of Newton Polynomial Evaluation

The polynomial in Newton form having the coefficients $a_{0},a_{1},\ldots,a_{n}$ and centers $x_{1},x_{2},\ldots,x_{n}$ is the polynomial $$ p(x) = a_{0} + a_{1} (x-x_{1}) +a_{2} (x-x_{1})(x-x_{2}) + ...
5
votes
0answers
79 views

How to understand $G_{k_0}\bigcap R_n\subset D_n$?

$G\subset R^N$ is a measurable set, $mG< +\infty$ ,and $\varphi_n(x)\rightarrow \varphi(x)$ in measure. $\sigma\in R,k_0\in Z^+$,$f(x,u)$ meets the Caratheodory condition (for almost all of $x\in ...
5
votes
0answers
107 views

Gluing Together Borel Measures

Does anyone know a standard reference for the following, which I assume is true: X a topological space, $\{U_i\}$ an open cover, $\mu_i$ a collection of regular Borel measures agreeing on overlaps. ...