Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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2
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1answer
41 views

Is this equation valid $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$,?

While reading a script I found this equation: $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$ and i cannot figure out how the author did this. I'd appreciate a step-by step equation for this ...
0
votes
1answer
342 views

A change of variables in the euler equation

If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation: $$z^2w'' + \alpha zw' + \beta w = 0$$ where $w$ is a function of $z$ and ...
1
vote
1answer
238 views

Legendre's equation polynomial solution

This is a problem on analytic solutions of ordinarry differential equations. Any help will be greatly appreciated. Please, try to be as specific as possible as I don't handle this material very well ...
3
votes
2answers
386 views

Analysis proof for repeating digits of rational numbers

"Every rational number is either a terminating or repeating decimal". I knew there's a proof for this using number theory's theorems, but I wish to find a purely analysis proof, that is: the series ...
3
votes
2answers
53 views

Application of FTC and change of variable

Let $f:[0,1]\to \mathbb{R}$ be continuous such that $$\int_{0}^{1} f(xt)dt=0$$ for all $x \in [0,1]$. Show that $f(x)=0$ for all $x \in [0,1]$. Using the FTC and substitution: ...
3
votes
3answers
254 views

Show this sequence is equicontinuous

I'm stuck on an analysis problem to which I've reduced to the following, so some assumptions may be superfluous. Let $\{ f_n(x) \} \subset C(X,\mathbb{R}^{\geq0})$ (i.e. $f_n$ is continuous and ...
1
vote
2answers
146 views

Convergence of a particular double series

For the double series $$ \sum_{m,n=1}^{\infty} \frac{1}{(m+n)^p} , $$ I was wondering when it converges. I want to use double integrals to estimate it, but I don't know how to write the process ...
1
vote
1answer
522 views

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$. My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
3
votes
2answers
170 views

Elementary application of Brouwer's fixed point Theorem

A professor of mine has suggested to me to look at this theorem and to find a problem related to it to explain in a future class. I found an understandable proof in "Linear operators" by ...
3
votes
1answer
339 views

baby rudin, chapter 10, (differential forms) theorem 10.27

I'm having difficulties with the reasoning in the proof of theorem 10.27 (regarding integration over oriented simplexes). say ...
3
votes
2answers
640 views

Intermediate value and monotonic implies continuous?

$I$ is an interval, $I^0$ is the interior of $I$. Let $f:I\to\Bbb R$ be a funcion with intermediate value property on $I$, and $f(x)$ is monotonic on $I^0$. Does it follow that $f(x)$ is continuous ...
4
votes
1answer
79 views

Convergence in $L^2(\mathbb{R}_+)$

I'm struggling with the following question: Let $f \in L^2(\mathbb{R}_+)$ and let $r > 0$. Show that $\sum \limits_{n =0 }^{\infty} \frac{1}{r} \int_{[rn, r(n+1))} f(t)dt\textbf{1}_{[rn, ...
5
votes
3answers
142 views

Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$

Find $$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
2
votes
2answers
71 views

Compactness and continuity

I am having some trouble with this: Let $f : X \to Y$ be continuous and bijective with $X$ compact then $f^{-1} : Y \to X$ is continuous? I am not quite sure if this is true, if is not I cant find ...
2
votes
1answer
709 views

Integrating a composite function

There was a related question concerning the spectral decomposition of a nested function at [1]. Simply put, do the solutions to the integral of a composite function bear any resemblance to the ...
13
votes
4answers
616 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
1
vote
1answer
198 views

$C^k$ norm for functions of several variables

For a function $f$ of single variable, its $C^k$ norm can be of the form $$|f(x)|_{C^k}=\sup_{0\leq i\leq k}\sup_x|f^{i}(x)|.$$ What does the usual $C^k$ norm look like for a function of several ...
0
votes
1answer
70 views

limit of an integral and estimate the limit of integrand

I have $$\lim \limits_{k\rightarrow 0}\int \limits_{\mathbb{R}^3}f_{k}(x)g(x)\rightarrow 0.$$ Could you please tell me if it is possible to show $$\lim \limits_{k\rightarrow 0}f_{k}(x)\rightarrow 0$$ ...
2
votes
1answer
181 views

Sine not a Rational Function Spivak

This is Chapter 15 Question 31 in Spivak: a) Show sin is not a rational function. By definition of a rational function, a rational function cannot be $0$ at infinite points unless it is $0$ ...
1
vote
1answer
1k views

Composition of Riemann integrable functions

I know that if $f:[a,b]\to[m,M]$ is Riemann integrable and $g:[m,M]\to\mathbb{R}$ is continuous, then $g\circ f$ is also integrable on $[a,b]$. I'm trying to think about the following 3 cases: 1) $f$ ...
3
votes
1answer
134 views

Question on “Proving $f(x) = 0$ everywhere”

I've some questions on Brian M Scott's proof for Proving $f(x) = 0$ everywhere. Please do combine my thread with the original which might be easier to read. Thank you. Let $f:[0,1] \to\mathbb{R}$ ...
1
vote
1answer
660 views

What is the norm of this bounded linear functional?

Let $a$, $b$ be two arbitrary but fixed real numbers such that $a < b$, let $C[a,b]$ denote the normed space of all continuous real (or complex) valued functions defined on $[a,b]$ with the maximum ...
5
votes
2answers
242 views

Integral $\int_0^{\infty} \sin(x^2)/x^2\,dx$

Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$ I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed. Thanks.
3
votes
1answer
60 views

A question on immersions

I am facing the following problem: Let $\alpha:\mathbb{R}\rightarrow\mathbb{R}^2$ and $\beta:\mathbb{R}\rightarrow\mathbb{R}^2$ be $C^1$ curves with $\alpha(0)=(0,0)=\beta (0)$, such that $\alpha ...
4
votes
2answers
122 views

Prove existence positive integers $\epsilon <|h\sqrt{m}-k\sqrt{n}|<2\epsilon$

Given positive integers $h,k$ and $\epsilon>0$ show that there exist positive integers $m,n$ so that the inequality $\epsilon<|h\sqrt{m}-k\sqrt{n}|<2\epsilon$
2
votes
1answer
80 views

Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?

Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
7
votes
1answer
179 views

$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$

let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
7
votes
1answer
472 views

Wave Equation, Energy methods.

I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem: Theorem 5 (Uniqueness for wave equation). ...
3
votes
1answer
133 views

Versions of L'Hôpital's rule

I am familiar with the following version of L'Hôpital's rule: Let $f,g:I\to\Bbb R$ be differentiable on the interval $I$, further assume that $g'(x)\ne0$. Let $a\in I$ and assume the limit $\lim_{x\to ...
2
votes
2answers
127 views

nested compact set question

Suppose $A \subset \mathbb R^n$ is not compact. Show that there exists a sequence $F_1 \supset F_2\supset F_3\supset\cdots$ of closed sets such that $F_k \cap A\ne\emptyset$ for all $k$ and ...
1
vote
1answer
111 views

Why is the following function not càdlàg?

I have constructed the following function but I can't see why it is not càdlàg on $[0,1]$: $$f(x)=\begin{cases} 1, & ...
1
vote
1answer
704 views

What are the range and the norm of this bounded linear operator?

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
3
votes
1answer
119 views

A problem about $C^1$-convergence! (Elliptic theory)

Let a function $u:\overline\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies $$\Delta u+f(u)=0 \ \ \ \mbox{in} \ \ \Omega,$$ and consider ...
3
votes
1answer
44 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
4
votes
0answers
226 views

Disintegration of Measures

I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background. Let $(\Omega, ...
0
votes
2answers
143 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
5
votes
3answers
106 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
1
vote
1answer
238 views

How to find the range and inverse of this linear operator?

Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
7
votes
0answers
177 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
4
votes
0answers
79 views

Is the graph of every real function a null set? [duplicate]

This question popped to my mind during an analysis lecture: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a (general) function. Is there an $N\subset \mathbb{R}^2$ with $\lambda^2(N)=0$, such that ...
0
votes
0answers
77 views

Uniform best approximation in Chebyshev/Haar systems and the necessity of compactness of the function domain.

A great deal of Chebyshev/Haar systems are given for intervals $]-\infty,\infty[$, $[0,\infty[$ and other noncompact subsets of $\mathbb{R}$. Nonetheless, the theory of uniform best approximations in ...
3
votes
1answer
61 views

The area of the set in which a polynomial is “small”

Prove that there exist a constant $C$ such that for every monic polynomial $P$, the area of the set $A=\{x : |P(x)|<1\}$ is at most $C$. Remarks: This puzzle holds for both the real and the ...
1
vote
1answer
34 views

Help with limit $\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt$..

Let $D\subseteq \mathbb R^2$ be an open set and $P:D\rightarrow \mathbb R$ continuous. For $y$ fixed how to evaluate, $$\displaystyle\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt?$$ I know the ...
1
vote
1answer
664 views

About an extension of Riesz' Lemma for normed spaces

The Riesz' Lemma is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for ...
0
votes
1answer
262 views

Chebyshev rational approximations to $\cos x$

How can we construct all the Chebyshev rational approximations of degree $3$ for $f(x) = \cos(x)$. So, I note that we first get the Taylor series of $\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} ...
0
votes
1answer
257 views

How to define limsup of a function

Let $f: [0, \infty) \rightarrow \mathbb R$. What is the definition of $$ \limsup_{x \rightarrow a} f(x)$$ for $a \in [0, \infty)$?
3
votes
1answer
341 views

Unions of disjoint open sets.

Let $X$ be a compact metric space (hence separable) and $\mu$ a Borel probability measure. Given an open set $A$ and $r,\epsilon>0$ $\ $does there exist a finite set of disjoint open balls ...
1
vote
0answers
414 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
5
votes
2answers
473 views

characteristic curves for second-order equations

Reading about characteristic curves for second-order equations, in particular semi-linear equations of second order with two independent variables: ...
5
votes
0answers
102 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...