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
392 views

Help me correct these properties of : $f_{n}(x)= nx(1-x)^{n}$? Is there maybe a typo in the sequence?

Examine the sequence of functions $(f_n)_{n\in \mathbb{N}}$ on $x\in[0,1]$: $$f_{n}(x)= nx(1-x)^{n}$$ Does $(f_n)_{n\in \mathbb{N}}$ converge pointwise or uniform? I will show that it does ...
5
votes
2answers
264 views

Is my Fourier Series computation done correctly?

See my fourier series calculation of this function if you please! $ f(t)=\left\{\begin{array}{ll} 0, & \text{for } \ -\pi<t<0 \\ 1, & \text{for } \ 0 < t < ...
1
vote
2answers
69 views

Evaluating the integral $\int\int_G \frac{\ln{(x^2+y^2)}}{x^2+y^2} dx dy$

I need to evaluate the following integral: $$\int\int_G \frac{\ln{(x^2+y^2)}}{x^2+y^2} dx dy$$ Here $G=\{(x,y)\in \mathbb{R}^2: 1 \leq x^2+y^2 \leq e^2\}$. I tried to use parametric transformations ...
1
vote
2answers
78 views

Real analysis question on continuity

Let $f$ be a functions from reals to reals given by the following rule: $f$ = $\sin x$ if $x \in \mathbb{Q}$ and $f$ = $0$ if $x \in \mathbb{R} - \mathbb{Q}$. At what points if $f$ continous? I ...
1
vote
1answer
206 views

Lipschitz condition on a second order nonlinear ODE?

Preliminaries: Let the matrix norm be $$\sqrt{\sum_{j=1}^n\sum_{i=1}^n a_{ij}^2}=||\mathbf A||.$$ I am trying to prove uniqueness and existence of a second order nonlinear ODE (Ordinary Differential ...
2
votes
1answer
72 views

How does one show that the definite Riemann integral of a function is $ > 0 $?

Let $ f \in R[a,b] $, where $ f \ge 0 $, and suppose that $ f(x) > 0 $ for a point of continuity $ x $ of $ f $. Then one can show that $$ \int_{a}^{b} f(x) ~d{x} > 0. $$ One can construct an ...
0
votes
2answers
51 views

Example of a function on non-compact metric space

I am looking for a sequence of functions with following property. a. Continuous function on non-compact metric space that is monotone and point-wise convergent but not uniform convergent. Is it ...
2
votes
1answer
95 views

How does one apply the Bessel and the Parseval equality to a function?

Given the system: $$\{\frac{1}{\sqrt{2\pi}}e^{int} \}_{n\in \mathbb{Z} } \ \text{in} \ C[0,2\pi]$$ How does one apply the Bessel and the Parseval equality to the functions $f(t) = t $ and $g(t)=t^2$ ...
4
votes
1answer
223 views

A function that has a derivative but is not integrable

How is it possible that the function $F(x)$ defined by : $$ F(x)=\left\{\begin{array}{ll} x\sqrt{x}\sin\frac{1}{x}, & x> 0 \\ 0, & x=0\end{array}\right. $$ $$ ...
1
vote
1answer
68 views

Calculate the sequence $\sum_{i=1}^n\frac{1}{f'(x_i)}=n$

let $f:[0,1] \mapsto\mathbb R$ be differentiable and $f(0)=0 $ $f(1)=1 $how prove that $$\forall n\in \mathbb N\ \exists x_1,x_2,\ldots,x_n\in[0,1] \text{ such that} ...
1
vote
2answers
85 views

Range of possible values for nth root of natural number n

I'm interested in finding the range of possible values for the $n$th root of a natural number $n \in \mathbb{N}$. Right now, my intuition is telling me that $\forall n \in \mathbb{N}, 1 \leq ...
1
vote
1answer
51 views

isometric homeomorphism on $\mathbb{R}^2$

Let $f:\mathbb{C}\to\mathbb{C}$ be an isometric homeomorphism: $$(\forall x,y\in\mathbb{C})(d\left( f(x),f(y)\right) =d(x,y))$$ Let $\Delta$ be a triangle or the interior of a triangle. What is ...
3
votes
2answers
242 views

Inverse Function Theorem/ Polynomial

I was thinking about this after I read about Jacobian conjecture. But I can't see what I did wrong? Maybe you can help me. Let $F: \mathbb{C}^n \to \mathbb{C}^n$ be of the form $F(x_1, \dots, x_n)= ...
1
vote
2answers
60 views

Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ ,does $f$ has the intermediate value property on $[0,\infty)$?

Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ does $f$ has the intermediate value property on $[0,\infty)$? Attempts: In $\mathbb{R}$, if $f$ is continuous ...
1
vote
1answer
544 views

Proof that a continuous function is bounded below

I have this question: Assuming the theorem that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds, prove that if $f\colon\mathbb R\to\mathbb R$ ...
0
votes
1answer
130 views

How to show that f in $C[0,2\pi]$ is continuous under “shifting” to $[\delta, 2\pi-\delta]$?

$f\in C[0,2\pi]$ and $f_\delta := f$ in $[\delta , 2\pi - \delta],f_{\delta}(0)=f_{\delta}(2\pi)=\frac{1}{2}(f(0)+f(2\pi))$ and $f_\delta$ is linear in $[0,\delta]$ and also in $[2\pi - \delta , ...
9
votes
2answers
80 views

Uniqueness of solution for a functional equation

Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that $$ f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a]. $$ ...
6
votes
2answers
91 views

A problem of evaluating an integral

How to prove that $$\int_{0}^{1}(1+x^n)^{-1-\frac{1}{n}}dx=2^{-\frac{1}{n}}$$ I have tried letting $t=x^n$,and then convert it into a beta function, but I failed. Is there any hints or solutions? ...
6
votes
1answer
314 views

Interior Sphere Condition

Let $\Omega\subset\mathbb{R}^N$ be a bounded open set. We say that $\Omega$ satisfies the interior sphere condition (ISC), if for all $y\in\partial\Omega$ there is $x\in\Omega$ and a open ball ...
1
vote
0answers
46 views

Orthnormalization of the system $(1,t,t^2) $ in $C[0,2\pi]$

Is the orthonormalization of the system: $${1,t,t^2}$$ in $C[0,2\pi]$ with $(f,g)=\int fg $ as scalarproduct given by : $$v_1= 1 $$ $$v_2 = t- \frac{\int_0^{2\pi} 1\cdot t \, dt ...
0
votes
1answer
147 views

On the convergence of a specific sequence of integrable functions

Let $\{f_n\}$ a sequence of measurable non-negative functions on $\mathbb{R}$ converging point-wise on $\mathbb{R}$ to $f$, and let $f$ integrable over $\mathbb{R}$. If $\displaystyle ...
1
vote
3answers
671 views

Show that there exists a positive real number $x$ such that $x^3 = 5$.

Here is what I've done so far: [First, want to show $b = 5$ is an upper-bound of $S$.] So, let: $$S = \{x \in \Bbb R : x \gt 0, x^3 \le 5\}, S \neq \emptyset$$ Assume that $b = 5$ is not an ...
6
votes
2answers
327 views

strange metric $d(x,y) = ||x|| + ||y||$ if $x\ne y$, $d(x,y) = 0$ if $x = y$.

Let $d : \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty]$ be defined by $$ d(x,y) = \left\{ \begin{array}{ll} 0 & : ~ x = y \\ ||x|| + ||y|| & : ~ x \ne y \end{array} \right. $$ where ...
1
vote
2answers
129 views

how prove $x_n=(1+a_{n})^\frac1n $ is convergent? such that {$a_n$} satisfied in following conditions

assume {$a_n$}$_{n=1}^\infty$ ,$a_n$ is none negative and real sequence that satisfied :$$1+a_{m+n}\leq (1+a_{m})(1+a_{n}) ,\quad m,n\in\mathbb N$$ how prove $x_n=(1+a_{n})^\frac1n $ is convergent? ...
1
vote
1answer
108 views

How to find the best polynomial approximation of order 2 of $f(t)=e^t$ in $(C[0,2\pi], ||.||) $

How does one find the best polynomial approximation of order 2 of $f(t)=e^t$ in $(C[0,2\pi], ||.||) $ What I have tried: If one orthonormalizes $1,t,t^2...$ $C[0,2\pi]$ one will get the legendre ...
0
votes
1answer
379 views

Space of piecewise continuous functions on $[0,1]$ is not complete under 2 norm

How to understand that $\check{C}[0,1]$ is not complete under the 2 norm and no Hilbertspace under $(f,g)=\int f\overline{g}$ Suppose $\check{C}[0,1]$ being the space of piecewise continuous ...
4
votes
2answers
440 views

Where to go after Advanced Calculus 2?

I will be finishing up Advanced Calculus 2 soon and I would like to continue self studying Analysis. I want to learn Real and Complex Analysis, Measure Theory and all that other good stuff. but I am ...
3
votes
3answers
48 views

Inequality of real numbers

Is it true that for $a,b \in \mathbb{R}$ and $p\geq 1$ (or $p\geq 2)$ there exists a constant $C>0$ independent of $a,b$ of course, such that: $(a-b)^{p-2} ab \leq C (a-b)^p$ Thanks a lot! :)
2
votes
2answers
225 views

Uniqueness of Bounded Solutions to Dirichlet's Problem in the Half-Space

Title basically says everything. Prove that if $u\in C^{2}(\mathbb{R}^{n}_{+})\cap C(\bar{\mathbb{R}^{n}_{+}})$ is a bounded solution of the BVP $$\left\{\begin{array} -\Delta ...
0
votes
1answer
64 views

Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
1
vote
3answers
83 views

limit of $\frac{e^{2x}-x^2+x}{\cos(x)-1}$ as $x\to 0$.

The title said it, what is: $$ \lim_{x\to 0} \frac{e^{2x}-x^2+x}{\cos(x)-1} = ~? $$ If I evaluate the term I get $1/0$, by looking at a graph I see that it goes to $-\infty$, but I don't now how ...
1
vote
0answers
79 views

How to argue this “standard elliptic estimates” for a convergence?

I am estudying the paper: INEQUALITIES FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH APPLICATIONS TO UNBOUNDED DOMAINS I - H. BERESTYCKI, L. A. CAFFARELLI, AND L. NIRENBERG At the page $482$, we have a ...
3
votes
2answers
503 views

Space of probability measures “complete”? (In the other sense)

I want to consider a space of probability measures on some set $X$, such that the space of measures is complete, not in the sense of complete probability measures (though probably that too), but as in ...
4
votes
1answer
80 views

If $f:\mathbb R\to\mathbb R$ is continuous and $f^3(x)=x$, then $ f(x)=x$ [duplicate]

Possible Duplicate: 3rd iterate of a continuous function equals identity function Assume $f:\mathbb R\to\mathbb R$ is continuous and $f^3(x)=x $ $\forall x$. How can I prove that$$\forall ...
0
votes
1answer
23 views

On the existence of functions with a particular convergence

Is the following scenario possible? Provide an example or argue why not. Let $\{f_n\}_{n=1}^{\infty}$ be measurable non-negative functions on $[0,1]$ converging to $f(x)$ pointwise Lebesgue-almsot ...
0
votes
1answer
216 views

Space of continuous functions under the 2-norm is dense in the space of the piecewise continuous functions

How can one show that the space $C[a,b]$ (space of continuous functions on the interval a to b) is dense in $\check{C}[a,b]$ (space of piecewise continuous functions from a to b) under the norm ...
0
votes
3answers
206 views

How could you identify a periodic function with a function on a circle

Guess the headline already said everything. If I have a periodic function, for example on the real line, how could it be identified with a function, say for example on the unit circle?
2
votes
2answers
2k views

Does Riemann integrable imply Lebesgue integrable?

Suppose a definite integral exists in the Riemann sense. Does that mean the integral exists as a Lebesgue integral, and do we get the same result either way?
1
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0answers
498 views

How does one find the Fourier coefficients of a piecewise continuous function?

How does one find the Fourier coefficients of the following functions $f:[-\pi,\pi]\rightarrow \mathbb{R}$ $f(t)=t$ for $-\pi < t < \pi$ and $f(-\pi)=f(\pi)=0$ $f(t)=-1$ for $-\pi < t < ...
1
vote
2answers
147 views

$\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$

Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
2
votes
2answers
639 views

continuous and strictly increasing implies differentiable

I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable. It may be because there are no bumps like in the ...
1
vote
1answer
79 views

Uniformly compact on a set?

I am wondering if anyone can provide me with the definition of a uniformly compact mapping on a set. The mapping is defined as a ``point-to-set'' map. I cannot seem to find any literature discussing ...
0
votes
1answer
115 views

Convergence of series with modified denominator

Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$. Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges, and prove or ...
1
vote
1answer
50 views

A problem about Convolution

If $\phi\in C_0^{\infty}(\mathbb{R}^N)$ and $\psi\in L_{loc}^1(\mathbb{R}^N)$ is defined by $\psi(x)=|x|^{2-N}$, $N\geq 3$ , does $\phi\star\psi$ is in the Schwartz space? Note: $\star$ stands for ...
0
votes
1answer
46 views

Unique point in a Convex set

If C is a convex set in a plane how can I show that there is a unique point M such that d(A,M)=d(A,C) where A is an arbitrary point.
1
vote
1answer
322 views

What are the differences between differential and gradient?

As far as i know, both differential and gradient are vectors where their dot product with a unit vector give directional derivative with the direction of the unit vector. So what are the differences?
1
vote
1answer
47 views

Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$?

It is possible to define a cozero-set of $X$ to be the cozero-set of some $f\in C^{*}(X)$, in fact; Every cozero-set in $X$ is the cozero-set of a function taking values in $[0, 1]$. $proof$: ...
8
votes
1answer
170 views

A functional equation problem

Let $f$ be a function which maps $\mathbb{Q}^{+}\to\mathbb{Q}^{+}$. And it satisfies $$ \left\{ \begin{array}{l} f(x)+f\left(\frac{1}{x}\right)=1\\ f(2x)=2f(f(x)) \end{array}\right. $$ Show that ...
1
vote
0answers
75 views

Finding a bound on $f'$

$f$ is an entire function such that $|f(z+w)| \leq |f(z)| + |f(w)|$ for any $z, w $ in $\mathbb{C}$. I need to show that $f(z) = az +b$ for some complex numbers $a$ and $b$. So, it suffices to show ...
1
vote
3answers
38 views

On building a subsequence inductively

Let $(s_n)$ be a sequence of real numbers and suppose that lim sup $ s_n =+ \infty$. I want to inductively build a subsequence $(s_{n_k})$ in $(s_n)$ such that lim $s_{n_k}=+\infty$. What would be my ...