1
vote
0answers
24 views

Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and it's corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
0
votes
0answers
15 views

A class of function to study Fourier analysis, which is a subset of BV functions.

In Fourier analysis, while talking about pointwise convergence, we generally start with the class of functions called, BV functions (functions of bounded variation), which have a finite total ...
4
votes
2answers
80 views

Bartle vs Rudin, which one is better for real analysis?

I'm in high school and I want to study real analysis, and I can choose between The elements of real analysis by Robert G. Bartle and Principles of mathematical analysis by Walter Rudin, so, from the ...
1
vote
1answer
82 views

Relation between the covers by sets of small diameter and the size of uniformly separated sets

Sorry I didn't find a better title. Here is the problem and my solution so far, I'd appreciate if someone could told me if is correct and for the last point, which at first sight seems to be ...
1
vote
1answer
49 views

$\sin(x)$ is asymptotically equal to $x+5x^3$

Here is my question: I've never seen before this kind of fact underlined about asymptotic equalities (and why we keep only one term in these equalities) and I'm looking for reference. Here is an ...
5
votes
1answer
205 views

A question about a mathematical analysis book

I am a newcomer to Analysis. All knowledge I know about "Analysis" are differentials,limit and integration (basically, what we have been taught in high school) I am studying Principles of ...
1
vote
0answers
46 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
1
vote
2answers
25 views

Verification of sequence result

Is it true that if a real sequence $\{x_n\}_1^\infty$ has an infimum but no convergent subsequences then the infimum must be the minimum as well? Secondly, can it be proved that the sequence defined ...
1
vote
1answer
82 views

Book recommendations for these types of math?

I'm planning to write a math olympiad in a couple of months (4-5), and am just really trying to get the preparation in. I'm a fairly good math student (did ok in math, not an A+, but I got an A so my ...
1
vote
0answers
36 views

Expected values of continuous and bounded functions are equal then random variables are equal, too.

I have seen several of reasoning based on the following fact: Real random variables $X, Y$ in $\mathbb{R}^n$ are equal almost surely if and only if $\mathbb{E}g(X)f(X) = \mathbb{E} g(X)f(Y)$ for ...
0
votes
1answer
27 views

Verification of extension result for Lipschitz functions

does anyone know the following result? If it holds in this form and any source which presents it? Thanks a lot. Consider metric space $(X,d_{X})$. Let $f:A \subset (X,d_{X}) \rightarrow \mathbb{R}$ ...
5
votes
0answers
108 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
2
votes
2answers
109 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
0
votes
0answers
26 views

Examples of elements in the Dirichlet space

By Dirichlet space we mean $\{F\in C_{0}[0,1]:\text{ there exists }f\in L^{2}[0,1]\text{ with }F(t)=\int_0^t f(x) \, dx$, $\forall t\in [0,1]\}$. The more the better. Any famous examples? Using the ...
4
votes
1answer
84 views

Proof for and Intuition behind Taylor's Theorem

I notice that multiple versions of a theorem are called Taylor (univariate/multivariate, approximate/exact). But I do not find it trivial to infer proof of one version from the rest. So looking for a ...
1
vote
1answer
103 views

Measure Theory Book

What book should I use for measure theory?I have solved Rudin's Principle Of mathematical analysis up to chapter 7.Some people advised me to use Real and complex analysis by Rudin, while other said it ...
2
votes
0answers
56 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
5
votes
0answers
40 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
3
votes
1answer
62 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
1
vote
1answer
59 views

Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
5
votes
1answer
69 views

If $f: A\to\mathbb R$ one-to-one but not monotone, there exist $x,y,z\in A$ with $x<y<z$ such that $f(x) < f(y)$ and $f(y) > f(z)$ (wlog)

The following result is part of the folklore, but I'd like to have a standard reference for something that I am writing: If $A \subseteq \mathbb R$ and $f: A \to \mathbb R$ is one-to-one but not ...
1
vote
1answer
144 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. ...
0
votes
2answers
217 views

Where can I find SOLUTIONS to real analysis problems? [closed]

I'm specifically interested in problem sets in Real Analysis that have solutions. I have a few books on it, but I'd like to compare my solutions with some given answers in a lot of cases to ensure ...
0
votes
1answer
32 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
1
vote
2answers
31 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
2
votes
3answers
92 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
5
votes
1answer
67 views

Limit everywhere, limit function is continuous, specific proof.

Suppose $f:[a,b] \to R$ is a function such that $\lim_{t\to x} f(t) = g(x)$ exists $\forall x \in [a,b]$. It can be shown that $g(x)$ is a continuous function. I seem to remember that there was a ...
1
vote
1answer
44 views

Where can I find introductory video lectures about calculus and analysis?

I am having calculus classes that are titled as Calculus for Mathematicians, for the rest of the students who are studying calculus, they use Stewart's book. In our classes, we're having something ...
1
vote
3answers
112 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
0answers
83 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
0
votes
0answers
26 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad ...
3
votes
1answer
60 views

Is there a garden of derivatives?

I've found a book called A Garden of Integrals, in which the author shows the evolution of the concept of Integral. I follow AnalysisFact on Twitter, some days ago, they posted the following: The ...
0
votes
0answers
23 views

Nature of Hessian of a function of a matrix

If input to a differentiable function is a matrix, what is the nature of Hessian of the function? Is it a tensor or something? This is a simple question, but I guess I am not sure where refer to, to ...
1
vote
1answer
45 views

Looking for “explicit” integrals solvable using lebesgue integration theory

I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main ...
0
votes
1answer
37 views

Introduction to bump functions/ mollifiers

I want to introduce a small perturbation into a particular point of a smooth function, and then, use bumo functions to smooth out the perturbed function. Could any of you recommend a good ...
1
vote
2answers
55 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
2
votes
1answer
44 views

Is there a general algorithm to solve computable integral equation?

Hilbert's tenth problem ask for the general algorithm(finite number of operation) to solve of all Diophantine problems.Today, it is known that no such algorithm exists in the general case. What ...
14
votes
2answers
357 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
4
votes
0answers
33 views

Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
1
vote
1answer
137 views

Real analysis text books

I'm sure his has been asked before but can someone recommend a real analysis text book with lots of worked examples practice questions and solutions? The sort of textbooks I've used and really ...
0
votes
0answers
19 views

reference for existence and blow up results in transport-like PDEs (Vlasov equation)

I'm looking for references to results regarding maximal time existence of solutions of a certain transport-like PDE, more precisely this one: $$ \partial_t f + v \cdot \nabla_x f + E(f,t,x) \cdot ...
4
votes
4answers
139 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
1
vote
1answer
36 views

Kolmogorov's Existence Theorem

My analysis professor told us to take the following theorem for granted in order to prove other results, but I would like to see a proof of it, since I think it will be beneficial. Here is the ...
1
vote
1answer
78 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
0
votes
1answer
30 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
5
votes
1answer
74 views

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
1
vote
0answers
68 views

Cannot find a good intermediate real analysis textbook with enough examples/solutions - Any suggestion?

I am currently taking an intermediate real analysis course, one that is between an undergraduate real analysis course and a graduate real analysis course in terms of difficulty. However, I have not ...
3
votes
0answers
105 views

Which book is better, Rudin or Ahlfors? [closed]

As a beginner, with no history in analysis, what book is better for self teaching; Rudin or Ahlfors? Thanks!
1
vote
0answers
20 views

smoothnes of rotationally symmetric functions

Consider a smooth function $f : [0,\infty[ \to \mathbb R$. This function induces a rotationally symmetric function $F : \mathbb R^2 \to \mathbb R$ via $F(x,y) = f(||x,y||)$. It should be correct ...
7
votes
1answer
147 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...