5
votes
0answers
24 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
56 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
47 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
64 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
72 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier stuff. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see ...
0
votes
2answers
127 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
26 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
24 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
90 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
65 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
41 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 ...
0
votes
3answers
75 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
56 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
59 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
15 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
39 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
35 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
1answer
37 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
39 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
320 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
101 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
17 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
137 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
31 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
75 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
28 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
69 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
60 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
94 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
18 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
132 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. ...
0
votes
1answer
28 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
votes
1answer
32 views

How do i prove “change of variables”?

To be honest, I don't get the proof in Folland, "Real Analysis", p.74. Let $\|\cdot\|$ be the max norm on $\mathbb{R}^n$. Let $\Omega$ be open in $\mathbb{R}^n$. Let $G:\Omega \rightarrow ...
5
votes
1answer
116 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
6
votes
1answer
134 views

Is it possible to extend a $C^1$-function smoothly from any Lipschitz domain?

If $\Omega$ is a cube in $\mathbb{R}^n$ and $f\in C^1(\overline\Omega)$. By reflection one can extend such a function to all of $\mathbb{R}^n$ and the extenstion is in $C^1(\mathbb{R}^n)$. If ...
0
votes
0answers
34 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
0
votes
1answer
100 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
0
votes
2answers
47 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
0
votes
0answers
9 views

analytic property of periodic properties

Can any one suggest me some books in which I can see the analysis of periodic functions? I dont have any constrain in the domain or codomain. For example this book ...
0
votes
1answer
38 views

Generalization of the Riemann integral to functions on the sphere

I want to do something a bit strange: define the Riemann/Darboux integral for bounded real functions defined on the n-dimensional sphere. The catch: I cannot use Lebesgue integration theory to do ...
3
votes
2answers
203 views

Is there another Analysis book that is based on the Cartesian space $\Bbb R^p$

I am in the middle of a slightly ambitious attempt to learn Analysis on my own. I skimmed through Rudin(Baby), Chapman Pugh, William Wade, Stephen Abbott and Strichartz and ended up preferring the ...
5
votes
2answers
303 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
5
votes
4answers
776 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
2
votes
1answer
38 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
3
votes
1answer
87 views

Exponentiation of Real Numbers?

I'm looking to learn Real Analysis on my own. Am reading Elements of Real Analysis by Bartle. I came across this project which defines the powers of real numbers i.e. exponentiation. Firstly I am ...
1
vote
5answers
210 views

What books on analysis after someone has finished all 3 by Rudin?

What books on analysis would people recommend after someone has finished all three by Rudin (Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis)? I am looking for ...
1
vote
3answers
75 views

Recommend me a text or webpage introducting gamma function throughly

Till now, i have learned abstract Integration, all basic properties of the (n-dimensional) Lebesgue(-Stieltjes) measure and the lebesgue integral is an extension of Riemann integral. Here's an ...
2
votes
1answer
287 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...