Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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4
votes
2answers
282 views

Continuity of the Characteristic Function of a RV

Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say ...
3
votes
1answer
470 views

$\sum \frac{z^{2n}}{1-z^{n}}$ normally convergent in $\mathbb{E}$

I tried to solve this exercise (Remmert Theory of Complex Functions, p. 107, exercise 1 ), but I didn't get very far: Proposition: $$\sum \frac{z^{2n}}{1-z^{n}}$$ is normally convergent in ...
6
votes
2answers
159 views

Showing that a function is the sine function

How can I prove the following: If $ f: \mathbb{R} \rightarrow \mathbb{C} $ is a $2\pi$-periodic function of class $C^{\infty}$ such that $f'(0)=1$ and that for any $n\in \mathbb{N}, ...
2
votes
1answer
245 views

$f_{n}=\frac{1}{1+z^{n}}$ uniform convergence

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\s}{\sigma}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\F}{\mathbb{F}}$ I am trying to show ...
1
vote
2answers
120 views

Monotone nondecreasing homeo => Lipschitz?

If I have a monotone non-decreasing $f:[a,b]\rightarrow [c,d]$ which is also a homeomorphism, is it necessarily Lipschitz? If yes, what would be a good candidate for the Lipschitz constant? I'm doing ...
1
vote
0answers
55 views

Nonsingularity of a matrix

I have a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ that is homogeneous of order one: $g(\lambda x) = \lambda g(x)$. Let $(x_1,...,x_n) \circ (y_1,...,y_n) = (x_1 y_1,...,x_n y_n)$. Are ...
3
votes
1answer
584 views

Limit for gamma function

How can I prove that $$\displaystyle \Gamma(z)=\lim_{n \to \infty} \displaystyle \int_0^n \left( 1-\frac{t}{n}\right)^n t^{z-1}\ \text{d} t\;=\displaystyle \int_0^{\infty} e^{-t} t^{z-1}\ \text{d} ...
9
votes
0answers
744 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 ...
5
votes
1answer
239 views

Does for every continuous function $f:R \rightarrow R$ there exist a sequence of analytic functions convergent uniformly to $f$?

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $\mathbb{R}$-analytic iff for every $x_0 \in \mathbb{R} $ there exist $R>0$ and power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ ...
2
votes
1answer
1k views

Every harmonic function is the real part of a holomorphic function

Is there a way to show that every harmonic function is the real part of a holomorphic function without using integration equations if later theorems are allowed also?
3
votes
3answers
328 views

Locally Bounded Functional Equation $f(x+y) = f(x) + f(y)$ and Continuity

Let $f$ be a real-valued function on $\mathbb{R}$ s.t. $f(x+y) = f(x) + f(y)$ for all $x,y$ reals. Suppose there are reals $c$ and $M$ s.t. $|f(x)| \leq M $ for all $x$ in $[-c,c]$. Show that $f$ is ...
0
votes
1answer
49 views

Is a map that satisfies $|z||w|\langle T(z),T(w)\rangle = |T(z)||T(w)|\langle z,w\rangle $ and isn't the 0 map an injection?

Remmert page 15 chapter 0 it says that angle preserving mapping is R-linear and injective. We want to prove: Given $$T:\mathbb{C}\rightarrow \mathbb{C}$$ a $\mathbb{R}$ linear map which satisfies ...
1
vote
0answers
99 views

Showing inequalities in complex variables

Remmert chapter 1 page 13 and 14 Set $z:= x+iy, w:= u+iv$: $$\langle z,w\rangle^2+\langle iz,w\rangle^2 = Re(w\bar{z})^2+Re(iw\bar{z})^2=(ux+vy)^2+(uy-vx)^2= (x^2+y^2)(u^2+v^2)= |z|^2|w|^2$$ so ...
1
vote
0answers
95 views

C-linearity, conformity, bijectivity of a function

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\s}{\sigma}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\F}{\mathbb{F}}$ Remmert chapter 0 ...
2
votes
0answers
85 views

factoring infinite products of $q$-series with constant term equal to 1

I was thinking about the following infinite product: $$\prod_{n=0}^{\infty} \frac{ae^{-2n}+be^{-n}+c}{c}$$ The right way of generalizing it is to think in terms of $q$-Pochhammer symbols. If $r_{1}$ ...
4
votes
3answers
349 views

Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
3
votes
2answers
555 views

What are the consequences if Axiom of Infinity is negated?

What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ...
3
votes
1answer
352 views

Basic properties of the point-to-set distance function

Let $X$ be a normed vector space, $x\in X$ and $Z\subseteq X$. Then we define the point-to-set distance function as: $$ \|x\|_Z = \inf_{z\in Z} \| x-z\| $$ I use the notation $\|\cdot\|_Z$ for ...
0
votes
1answer
345 views

Rectifiable Sets

I'm trying to give an example of a bounded set of measure zero that is rectifiable and an example of a bounded set of measure zero that is not rectifiable. I can't think of 1 the top of my head, and ...
5
votes
2answers
205 views

Is this series convergent?

$a_n=(-1)^{k_n}\frac{1}{n}$ where $(k_n-1)^2<n\leq k_n^2$. Is the series $\sum_n a_n$ convergent? I tried with all the classical methods, but they seem to fail, any hint? EDIT: I had an idea: ...
1
vote
0answers
65 views

Fixed Point Theorem for Set-to-Set Mappings

Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$: $$ T:S\supseteq A \mapsto T(A)\subseteq S $$ Where $S$ ...
8
votes
7answers
865 views

Proving the positivity of a twice-differentiable real-valued function

This is a problem from Berkeley prelim exams, Spring '99 Suppose that $ f $ is a twice differentiable real-valued function on $\mathbb{R}$ such that $ f(0) = 0 $, $ f'(0) > 0 $, and $ ...
2
votes
1answer
155 views

What locally integrable function $f$ satisfies $\int_a^ b f(x) \phi'(x)dx=0 $ for each $\phi \in C_0^\infty(a,b)$

Let $f:(a,b) \rightarrow \mathbb{R}$ be locally integrable and such that $$\int_a^ b f(x) \phi'(x)dx=0 \textrm{ for each } \phi \in C_0^\infty(a,b).$$ How to show, without help of distribution ...
3
votes
1answer
369 views

Minimize distance between 2 functions

Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably ...
14
votes
3answers
740 views

Intersection between orthogonal complement of a subspace and a set

Consider the normed vector space $E=\mathbb{R}^n$. Define $ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$. Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
2
votes
1answer
509 views

Implicit Function Theorem computation problem

Problem 1, page 78 of Munkres (Analysis on Manifolds): Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = ...
1
vote
1answer
622 views

Determinant of the “bordered” Hessian of a composition

Write $H_{f}$ for the Hessian of a real function $f:\mathbb{R}^n\mapsto \mathbb{R}$, and define the bordered Hessian as $$ H_{f} = \left(\begin{matrix}0 & \nabla f' \\ \nabla f & H ...
1
vote
0answers
74 views

Relationship between Number of circles required to surround a circles and the distance function?

In Why is a circle in a plane surrounded by 6 other circles, the implicit assupmtion is the distance is Euclidean, my question is: Are there any relation between the distance function being used and ...
8
votes
5answers
896 views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
0
votes
1answer
57 views

The meaning of the differential map between two functions space

... a continuous differential map $\dfrac{d}{dx} : C^k(\mathbb R)\rightarrow C^{k-1}(\mathbb R)$ ... I was wondering why a differential map could from $C^k(\mathbb R)$ to $C^{k-1}(\mathbb R)$, ...
1
vote
1answer
280 views

Problem involving a hyperplane and affine subspace II

I am trying to solve this little problem. Suppose you have a normed vector space $E$. Let $H$ be a hyperplane ( $H=\{x\in E: f(x)= \alpha\}$ for some linear functional $f$ and some real number ...
1
vote
2answers
147 views

help with the Riemann - Stieltjes

why can I say that $$ \int_0^a t^2 dF(t) = \int_0^a t^2 d(F(t) -1) $$ unfortunately my experience with the Riemann -Stieltjes is practically non existent, so for instance I do not understand, why ...
0
votes
1answer
164 views

Analytically solving limits

I read the theory of limits and i have some misunderstanding. For example we have simple limit expression: $$\lim _{x\rightarrow \infty}{\frac{1}{x}}$$ I see that this limit is 0 and if build graph ...
2
votes
1answer
99 views

Equality of integrals of differential forms

I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...
2
votes
2answers
313 views

About example of continuous function on $\mathbb{R}$ which cannot be uniformly approximated by polynomials? [duplicate]

Possible Duplicate: Weierstrass approximation does not hold on the entire Real Line If a function $f: \mathbb{R}\rightarrow \mathbb{R}$ is continuous then $f$ can be uniformly approximated ...
2
votes
2answers
160 views

Differentiablility of a function of two variables

Here is a problem from an old comprehensive exam that I am trying to solve Problem: let $f:\mathbb{R}^{2} \to \mathbb{R}$ be a function defined as follows: $f\left ( x,y \right )=\frac{\left ( ...
4
votes
1answer
182 views

The set of diffeomorphisms preserving some metric.

Let $M$ be a finite-dimensional, smooth manifold. Call a diffeomorphism $f : M \rightarrow M$ diagonalizable if there exists a Riemannian metric $g$ on $M$ such that $f : (M, g) \rightarrow (M, g)$ is ...
2
votes
1answer
654 views

Supremums of measurable functions

According to my textbook, supremums of measurable functions exist and are measurable. But what about the sequence of functions $f_n: [0, 1] \to \mathbb{R}$ given by $f_n = n$? I don't think this ...
5
votes
1answer
692 views

How smooth is a smooth function?

Let's say a smooth function is a $\mathcal{C}^\infty$ function on $\mathbb{R}$. Some smooth functions are not analytic, the most notorious example being the bump functions. A non-analytic ...
2
votes
1answer
473 views

Definition of Borel sets

MathWorld says: Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable unions and intersections. Formally, the class of Borel ...
1
vote
1answer
81 views

How to solve quadratic over square root of quartic equals constant?

We need to solve the following equation for l. $$\frac{n_1 - \ell n_2 + \ell ^2 n_3}{\sqrt{d_1 - \ell d_2 + \ell ^2d_3 - \ell ^3d_4 + \ell ^4d_5}} - \cos{a_0} = 0$$ We have already tried ...
4
votes
1answer
155 views

Is this set closed?

$X=(C[0,1],\rho_\infty)$ where $\rho_\infty$ is the uniform norm. $M\in(0,\infty)$, define $A=\{f\in X:f(0)=0, f\;\mathrm{differentiable\;on}\;(0,1)\;,|f^\prime(x)|\leq M\;\;\forall x\in(0,1)\}$. I ...
6
votes
2answers
1k views

How to prove convex+concave=affine?

Suppose $f:R^n\to R$ is both convex and concave, how to prove that $f$ is linear? or exactly speaking, $f$ is affine? I thought for the whole day, but I cannot figure it out. When I was working on ...
2
votes
0answers
249 views

Characterization of Asymptotic Stability via KL-class functions

Let us adopt the following definition of stability and asymptotic stability of a dynamical system of the form: $$ \dot{x}=f(x) $$ The trajectory of this system starting from the initial point ...
3
votes
1answer
320 views

Solving integral equation with Laplace's Transform.

I'm trying to prove the following $$\int\limits_0^\infty {\frac{{\cos tu}}{{{u^2} + 1}}\log udu} = - \frac{\pi }{2}\int\limits_0^\infty {\frac{{\sin tu}}{{{u^2} + 1}}du} $$ The original problem ...
2
votes
0answers
138 views

Help with understanding a proof in Fourier Analysis

I have a stack of lecture notes that I am currently going through to teach myself a little bit about Fourier Analysis. Now I struggle with the following Lemma, which is needed to talk about the ...
2
votes
0answers
74 views

(RESOLVED) Interpreting a holomorphic function

The equation for and electric field is given by $E=−∇ψ$ where $\psi$ is the potential, and in this case $ψ=−Q\ln r$ where $Q$ is just some constant. I have found its harmonic conjugate to be $−Qθ+c$ ...
4
votes
0answers
165 views

Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion ...
1
vote
0answers
88 views

Is the following function convex-\cap?

Let $p=(p_1,\ldots,p_n)$ be a given nondegenerate (i.e., all $p_i> 0$) probability distribution on $n$ points. Define the following function $$\Phi(b_1,\ldots,b_n)=\frac{\left(\sum_{k=1}^n b_k ...
0
votes
1answer
194 views

Harmonic conjugate

I have been asked the following question and would appreciate an explanation. Suppose we have to find an analytic function $F(z)$ where $z=x+iy\in \mathbb C$ and its real part is $g(x,y)$. Question: ...