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
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0answers
126 views

$\max_{y} \min_{x} f(x,y)$ as motif for exploring mathematics

It's been several years since my undergraduate math days, and I would like to spend a bit of time refreshing and then tackling a few things I never completely mastered. Rather than proceeding topic ...
2
votes
0answers
113 views

Analytical solution for an almost geometric series

Is there any way of solving explicitly the limit of the series $\sum_{n=0}^\infty q^n a^{p ^ n}$ where $0<p,q<1$ and $a>0$? The series is obviously convergent as $a^{p ^ n} < ...
3
votes
1answer
59 views

Functional disequality

Let $f \in C^{2}([a,b]) \ $, $f(a)= f(b) = 0 \ $, $f(x) > 0 \ \forall x \in (a,b) \ $, $f(x) + f(x)''>0 \ $. Then $b-a \ge \pi $. Any hint?
2
votes
1answer
125 views

Growth rate of analytical functions

Given any computable function $f(x)$, is there an algorithm to find a set of coefficients $a_n$, such that i) $g(x)=\sum_{n=1}^{n=\infty} a_nx^n$ converges for all $x>1$ ii) $g(x)$ eventually ...
1
vote
1answer
152 views

convergence / fixed point method

Any help with the following: Problem: Consider the fixed point problem: $x=f(x)$ and given: $x_{n+1}=\frac{n}{n+1}f\left ( x_{n} \right )$. If $x_{0}$ is a fixed point where $\left | f^{'}\left ( ...
2
votes
1answer
92 views

At which points is this function continuous?

At which points is the following function continuous? $$\begin{eqnarray*} f(x) = \begin{cases} 5x, &\text{if }x \in\mathbb Q, \\ x^2-6, &\text{if }x \notin\mathbb Q. \end{cases} ...
10
votes
3answers
795 views

What is the insight behind the Lebesgue integral?

Edit 3: OK, I had an insight, inspired in part by Ben-Blum Smith's comment, and the post he linked to. (I have no idea if this insight is right; it's barely a hunch, and that's why I'm not submitting ...
1
vote
1answer
88 views

Decreasing function made up of cubes, squares and floor function

Let $t$ and $r$ be two integers with $r\geq 1, t\geq \frac{r}{2}$. Put $$ f(r,t)=\lfloor 2(t^2+r)^{\frac{3}{2}}-(2t^3+3rt) \rfloor $$ (here $\lfloor x \rfloor$ denotes the floor of $x$, i.e. the ...
1
vote
2answers
90 views

Functions bounded by sum of its previous values

How to find or bound the fastest growing function $f(n)>0$, with $f'(n)>0$ such that $f(n+1)<\sum_{j=1}^n f(j)$ ? Is $\ln(f(n))\ll n$ necessary and sufficient?
3
votes
1answer
85 views

An inequality with radicals

If $s_{1}\ge t_{1}\ge t_{2}\ge s_{2}\ge0$, does one always have $(s_{1}-t_{1}+s_{2}+t_{2})^{1/2}\ge\sqrt{s_{1}}-\sqrt{t_{1}}+\sqrt{t_{2}}-\sqrt{s_{2}}$? Thanks a lot!
1
vote
1answer
354 views

Convex Conjugate of Absolute Norm

Let $f:\mathbb{R}\rightarrow[-\infty,\infty]$ be a continuous function. The convex conjugate of $f$ is: $$f^*(p) := \sup_{x\in\mathbb{R}}\{px-f(x)\}~.$$ Furthermore, let us define the subderivative ...
0
votes
2answers
154 views

What are eigenvalues of such an operator?

Let operator $A: L^2(0,2\pi) \rightarrow L^2(0,2\pi)$ be given by $$(Au)(x)=\sin x \int_{0}^{2\pi} u(y)\cos y \, dy$$ for $u\in L^2(0,2\pi), x\in [0,2\pi]$. What are eigenvalues of $A$? Thanks ...
4
votes
2answers
344 views

Extension of uniformly continuous function

I want to prove this: $f\in C((a,b))$ uniformly continuous. Then there exists $\tilde{f}\in C([a,b])$ extension of $f$. I took $x_n\rightarrow a$ and defined $\tilde{f}(a)=\mathrm{lim}\;f(x_n)$. I ...
1
vote
1answer
288 views

Maximum inequality

Can anyone show that ...
0
votes
2answers
288 views

Continuity Definition for Real Functions

Continuity is defined in my book Basic Analysis (by Lebl pg 86) like this: Let $S \subset \mathbb R$, $f : S\rightarrow \mathbb R$ be a function, and let $c \in S$ be a number. We say that $f$ is ...
1
vote
2answers
124 views

Continuous Functionals and Norms

In Luenberger Optimization book, pg. 40 upper semicontinuity for a functional is defined as "if given $\epsilon > 0$ there is a $\delta > 0$ s.t. $f(x) - f(x_0) < \epsilon$ for $||x-x_0|| ...
3
votes
1answer
344 views

Hardy's inequality again

How can I prove that the constant in classical Hardy's inequality is optimal? $$\int_0^{\infty}\left(\frac{1}{x}\int_0^xf(s)ds\right)^p dx\leq \left(\frac{p}{p-1}\right)^p\int_0^{\infty}(f(x))^pdx,$$ ...
1
vote
1answer
81 views

A simple question about the difference between directional derivatives and derivatives

As the topic, let $\mathbf{u}=u\hat{i}+v\hat{j}$. Why $$D_ug(0,0) = \lim_{h\rightarrow0}\frac{g(hu,hv)-g(0,0)}{h}$$ while $$Dg(0,0)= \lim_{(h,k)\rightarrow ...
1
vote
1answer
182 views

Existence of a measure

I need help in showing that if $\alpha$ and $\beta$ are measures defined on $\mathfrak{A}$, and $\beta \leqslant\alpha$ then there is a measure $\lambda$ on $\mathfrak{A}$ such that ...
0
votes
1answer
60 views

How to construct an increasing sequence of sets.

This is part of a bigger problem I'm working on. To construct a a decreasing sequence of sets, $A_{n}\supseteq A_{n+1}$, I did the following: Let $B=\cup_{n=1}^{\infty} B_n$ and set $A_n= B\setminus ...
4
votes
3answers
241 views

How many solutions has $z^\pi = 1$?

I know that for $z \in \mathbb C$ and some natural $n\geq 1$, the equation $z^n = 1$ has exactly $n$ solutions. But what if I say $n$ need not be natural, e.g. $$ z^\pi = 1.$$ I mean the equation ...
3
votes
4answers
168 views

simple real analysis question on integration

it is trivial that $\int_0^{2\pi} \cos(x)\,dx = 0$. Intuitively, it is clear that for a strictly decreasing positive function $f(x)$, $$ \int_0^{2\pi} f(x) \cos(x)\,dx \ge 0 $$ but I have no glue ...
0
votes
2answers
146 views

Maximum in a cell intersected with a sphere

I have a rectangular cuboid-shaped 3D "cell" with scalar values at each vertex $(v_1,\ldots,v_8)$. Within this cuboid I do tri-linear interpolation. What I want is the maximum value of that function ...
7
votes
2answers
340 views

Divergent Series

Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly): Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or ...
0
votes
1answer
1k views

Complex Analysis: Laurent Expansion for $\frac{1}{\sin(z)}$ on different annuli.

I've been working through Hilary Priestley's Book Complex Analysis (fantastic read) and have reached her discussion of the Laurent Expansion for holomorphic functions. Considering the function ...
2
votes
2answers
183 views

What is the limit of $\lim\limits_{n\rightarrow\infty}\frac{1}{n^4}\left(\sum_{k=1}^{n}\ k^2\int_{k}^{k+1}x\ln\big((x-k)(k+1-x)\big)dx\right)$

As the topic how to find the limit of $$\lim_{n\rightarrow\infty}\frac{1}{n^4}\left(\sum_{k=1}^{n}\ k^2\int_{k}^{k+1}x\ln\big((x-k)(k+1-x)\big)dx\right)\;.$$
8
votes
3answers
3k views

Difficulties with Chapter 2 in Rudin

I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems ...
1
vote
2answers
369 views

Analysis on Manifolds Munkres Integration

I needed help in showing that the set $R^{n-1} \times 0$ has measure zero in $R^n$. What I have so far: Let $\epsilon > 0$. If $i_1,\dots,i_{n-1}$ are integers, then define ...
2
votes
2answers
105 views

In such way some condition implies nonexistence of derivative?

Assume that a function $f:[0,1] \rightarrow \mathbb{R}$ is continuous. In what way condition $$\forall_{n\in \mathbb{N}} \forall_{0\leq x\leq 1-\frac{1}{n}} \exists_{0<h<1-x} \left| ...
2
votes
3answers
228 views

Intermediate growth rates

Is there any simple function/formula $f(n)$, which eventually dominates every $cn$ for every $c$, and is eventually dominated by $a \cdot n \cdot \ln^k(n)$ for every $a,k \in \mathbb{Z}$, where ...
0
votes
2answers
350 views

Why does the series $\sum_k\frac{(-1)^k}{k+x^2}$ converge uniformly but not absolutely?

Consider the series $\sum_k\frac{(-1)^k}{k+x^2}$. Why does it converge uniformly on $[0,\infty)$? and why doesn't it converge absolutely, always on $[0,\infty)$? The only thing that I noticed is that ...
4
votes
3answers
241 views

Borel Measure such that integrating a polynomial yields the derivative at a point

Does there exist a signed regular Borel measure such that $$ \int_0^1 p(x) d\mu(x) = p'(0) $$ for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
6
votes
5answers
2k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. ...
9
votes
0answers
528 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 ...
7
votes
3answers
446 views

Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$

In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
2
votes
3answers
263 views

How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
12
votes
1answer
693 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
4
votes
1answer
522 views

An existence of global solution of differential equation of first order

Let $f: (a,b) \times \mathbb{R} \rightarrow \mathbb{R}$ be of class $C^1$ in $D:=(a,b) \times \mathbb{R}$ and satisfies condition $$| f(t,x)| \leq A+B|x| \textrm{ for } (t,x) \in D,$$ where $A,B$ ...
1
vote
1answer
181 views

Problem with Picard Iteration

I have $ \frac{dy}{dx} = y^2, y(0) = y_0 $ I have solved this as $y = \frac{y_0}{1 - x y_0}$ Which has the Taylor expansion $ y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$ However, when I ...
2
votes
2answers
283 views

Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$

Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
2
votes
2answers
183 views

$f(x,y)$ in polar coordinates

So, I have $ f(x,y) = (x^2-y^2, 2xy) $, which is a local $\mathcal C^1$ isomorphism in $\mathbb R^2 \setminus \{(0,0)\}$. I have to write this function in polar coordinates: $$f(x,y) = ...
7
votes
3answers
552 views

Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
2
votes
1answer
203 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
0
votes
1answer
472 views

Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?

Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image? I know that this question is so connected with computer ...
2
votes
0answers
181 views

Solution to polynomial equations with non-radicals

For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial ...
0
votes
1answer
73 views

finding the function

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which is continuous and satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How ...
2
votes
1answer
182 views

Requirement of closed and bounded set $[a,b]$ in the Ascoli theorem

In Wikipedia, the Ascoli theorem requires the functions to be continuous on the closed and bounded interval $[a,b]$. However, in the proof given in the book "Theory of Ordinary Differential ...
4
votes
0answers
158 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
5
votes
1answer
186 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
1
vote
1answer
67 views

weak solutions need to have local integrability condition?

I am currently studying Poisson and Laplace equations. This is just a small question that has been causing me some confusion, and I would like some clarification before I resume my study. For example, ...