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).

learn more… | top users | synonyms (1)

15
votes
4answers
3k views

Difference between calculus and analysis

It's somthing I always want to figure out, when did calculus start to be extended to analysis(I reformulate the question, the previous one"where one can draw a line to distinguish calculus and ...
3
votes
1answer
382 views

Radius of convergence of the inverse of a power series

Let $a = \sum_k a_k X^k \in \mathbb C [\![ X ]\!]$ with $a_0 = 1$ and convergence radius $\rho_a > 0$. I want to show that the convergence radius of the inverse $b = \sum_k b_k X^k \in \mathbb C ...
2
votes
3answers
147 views

question on absolute continuous function.

I am given $f,f' \in L^1(\mathbb{R})$, and f is absolute continuous, I want to show that: $$\lim_{|x|\rightarrow \infty} f(x)= 0$$ Not sure how to show this, I know that $f(x)=\int_0^x f'(t) \, ...
15
votes
4answers
468 views

The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$

Does anyone know if this function has a name? I came up with it by looking at the power series for $e^z$, changing the summation to an integral, and substituting the gamma function for the factorial ...
3
votes
1answer
83 views

existence of a harmonic function

Let $\Omega\subset\mathbb R^n$ open, not bounded and $n\ge3$. Let $\partial\Omega$ bounded and regular concering the laplace operator. Given a continuous function ...
8
votes
5answers
1k views

Advice for benefits to directly use analysis textbook to replace calculus

Main purpose: For self-learning performance, neither for exam nor degree courses. Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer Question Description: I've been ...
0
votes
1answer
73 views

Ordering some functions without losing continuity

Let's consider the set $\mathbb R^{n}/S_n$, i.e. the quotient of $\mathbb R^{n}$ modulo permutations. An element $a \in \mathbb R^{n}/S_n$ is simply a $n$-tuple of real numbers (the order does not ...
6
votes
2answers
190 views

An inequality involving integrals

Let be $f:[0,1] \longrightarrow R $, $f$ is an integrable function such that: $$\int_{0}^{1} f(x) \space dx = \int_{0}^{1} xf(x) \space dx=1$$ I need to prove that: $$\int_{0}^{1} f^2(x) \space ...
3
votes
2answers
189 views

Infimum of a union

I have a set $X$ and a function \begin{equation} f: X \rightarrow \mathbb{R} \end{equation} and I am interested in the value \begin{equation} \inf\limits_{x \in X} f(x) \,. \end{equation} I can ...
1
vote
2answers
787 views

proving facts about $\alpha$-Hölder-continuous functions

I am studying myself some facts about $\alpha$-Hölder-continuous functions but I don't get any further by proving the following: $(1)$ $\forall\alpha\in ]0,1]$ is $C^{0,\alpha}$ dense in $C^0(D)$ ...
3
votes
2answers
64 views

analysis limit question

Let f be an integrable function on $\mathbb{R}$. Show that $\lim_{t\rightarrow 0} \int_{\mathbb{R}}|f(x + t) -f(x)|dx = 0$. I can make it work once it is shown to be true for $f\in C_c(\mathbb{R})$ ...
4
votes
1answer
87 views

Inequality between volume and its projections

Let $A \subset \mathbb{R}^3$ be connected and let's define $A_1, A_2, A_3 \subset \mathbb{R}^2$ as projections of $A$ onto three perpendicular (to each other) planes. Show that: $$|A| \le \sqrt{|A_1| ...
1
vote
2answers
382 views

An inequality problem. [duplicate]

Possible Duplicate: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$ In the condition $a,b \in[0,\infty)$, $1\le p<\infty$, How can I conclude this ...
-1
votes
0answers
187 views

In Hölder's inequality [duplicate]

Let $1<p<\infty$ and assume $f \in L^p$ and $g \in L^{p'}$. And $f \ge 0, g \ge 0$. Then I know Hölder's inequality can be implied. Then I want to know why $f(x)^p = g(x)^{p'}$ $\mu$-a.e. ...
4
votes
1answer
746 views

Generalization of Hölder's inequality

Assume $1<p_k< \infty$ for $k=1,\ldots,N$ , and $\displaystyle\sum^N_{k=1}\frac{1}{p_k} =1$. I want to prove that $$\left|\int_X f_1 f_2\cdots f_N\; d\mu \right| \le \lVert f_1\rVert_{p_1} ...
3
votes
0answers
92 views

What is this norm?

If $A(t)$ denotes for each fixed $t$ a (smooth) surface in $\mathbb{R}^n$, what is the norm on the space $$L^2\left(\cup_{t \in [0,T]} A(t)\times \{t\}\right)?$$ Is it $$\lVert f \rVert^2 = ...
4
votes
2answers
501 views

Interior, exterior and boundary of sets in $\mathbb R^2$

Can someone teach me how to find interior, exterior and boundary of these two sets in the plane, $\mathbb R^2$? The metric is $d_2 (x,y)=\sqrt{(x_1-y_1)^2 +(x_2-y_2)^2}$, where $x = (x_1, x_2)$ and $y ...
4
votes
1answer
229 views

Why it is sufficient to show $|f'(z)-1|<1$?

According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class ...
1
vote
2answers
167 views

A quick question on Baby Rudin Theorem 2.40: Every k-cell is compact.

I have a quick question on the excerpt of Theorem 2.40 of Baby Rudin. How would I get "If n is so large that $2^{-n}\delta<r$...."? I think that to get such an 'n' has something to do with exercise ...
2
votes
1answer
144 views

Assess the limit: $ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$

Compute the following limit: $$ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$$ I'm looking for an easy approach if possible.
5
votes
1answer
419 views

Cauchy-Product of non-absolutely convergent series

While grading some basic coursework on analysis, I read an argument, that a Cauchy product of two series that converge but not absolutely can never converge i.e. if $\sum a_n$, $\sum b_n$ converge but ...
2
votes
1answer
132 views

need help with real analysis question

I really need help with this question: Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover. Thanks a lot.
3
votes
1answer
110 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
2
votes
1answer
210 views

Asymptotic expansion of a special integral

I need an asymptotic expansion of J(n) $J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ Can anybody help to find the asymptotic analytically or at ...
5
votes
2answers
333 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
6
votes
2answers
191 views

To define a measure, is it sufficient to define how to integrate continuous function?

Let me make my question clear. I want to define a measure $\mu$ on a space $X$. But instead of telling you what value I assign for some subset of $X$ (measurable sets that form a $\sigma$-algebra), I ...
0
votes
2answers
106 views

How do I read this?

I received the following equation. It is supposed to contain clues to something I have to solve. I am not familiar with the math symbols used here. How do I read the following: ...
7
votes
1answer
298 views

Continuous root map of the coefficients of a polynomial

I have a set of polynomials $P_t(z)= z^n+ a_{n-1}(t)z^{n-1}+\cdots+ a_0(t)$ which depends on a real parameter $t \in [a,b]$ and where $a_{n-1}(t),\ldots, a_0(t)$ are real continuous functions. May I ...
10
votes
1answer
200 views

Every path has a simple “subpath”

I've been thinking about this for a while, and can't seem to find any way to do it despite the statement itself seeming obvious. The problem is: Let $f:[0,1] \to \mathbb{R}^n$ be a continuous ...
4
votes
1answer
137 views

How to find the function $f$ using this equation?

Assume that we have a function $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ and $D$ a constant. How can we solve the following equation for $f$: $$ \int^{b}_{x_2=a} \int^{b}_{x_1=a} ...
5
votes
1answer
306 views

Poisson summation formula and Schwartz functions

I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on ...
5
votes
1answer
147 views

How to solve $y=\frac{(x-\sin x)}{ (1-\cos x)}$

I only see numerical approaches to solve this equation. Is there an analytical solution to solve $x$ as a function of $y$ for the range $(0,2 \pi)$? If there is no solution, is it possible to proof ...
6
votes
1answer
189 views

Natural question about weak convergence.

Let $u_k, u \in H^{1}(\Omega)$ such that $u_k \rightharpoonup u$ (weak convergence) in $H^{1}(\Omega)$. Is true that $u_{k}^{+}\rightharpoonup u^{+}$ in $\{u\geqslant 0\}$? You can do hypothesis on ...
2
votes
2answers
189 views

Parabolic PDE local and global existence

If you have a local solution to a parabolic PDE (say we know it exists (weakly anyway) from time 0 to T), then if the solution is bounded in an appropriate way (in which norms?) then we can apparently ...
1
vote
1answer
85 views

Touch Typing Index - Speed and Accuracy

I am trying to determine the ability of my students to touch type. I have data on their speed (in seconds) and their accuracy (number of errors). I also know the number of words in the test (50 ...
5
votes
1answer
120 views

Is the functional $F(u) = \int_{\Omega} \langle A(x) \nabla u, \nabla u \rangle$ convex?

The functional \begin{equation} F(u) = \int_{\Omega} \langle A(x) \nabla u, \nabla u \rangle \end{equation} where $A$ is a symmetric matrix . You can assume $\Omega$ conviniente such that the ...
17
votes
2answers
748 views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
2
votes
1answer
257 views

Rudin's Principles of Analysis Theorem 1.11

I'm having a problem with this theorem. What if set $B$ is all $x$ such that $\sqrt{2} < x \le 2$, and $S$ is the set of all $y$ such that $\sqrt{2} < y \le 3$. $\sup L = \sqrt{2}$, which does ...
5
votes
1answer
207 views

Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs

The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + ...
2
votes
0answers
339 views

Arzelà - Ascoli Theorem for an arbitrary family of functions

I have just completed an exercise in Abott's Understanding analysis concerning the Arzelà - Ascoli theorem for functions on $\Bbb{R}$. The statement of the exercise is as follows: Let $\{f_n\}$ be ...
3
votes
0answers
345 views

weak lower semicontinuity of some functionals.

Let $(\nu_{j} - \phi)$ be is a bounded sequence in $W^{1,p}_{0}(\Omega)$. By reflexivity, there is a function $u \in W^{1,p}(\Omega)$ such that, up to a subsequence $$ \nu_{j} \rightarrow u \ ...
1
vote
1answer
721 views

A function and its Fourier transform cannot both be compactly supported

I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the ...
3
votes
1answer
462 views

locally lipschitz implies lipschitz

Suppose a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is locally Lipschitz. Prove $f$ is Lipschitz on $[a,b]$. Here is what I have so far: Let $[a, b]$ be some closed, bounded interval. Since f is ...
1
vote
1answer
132 views

Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.

Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed. What I know: If $L$ is a finite dimensional subspace, then $L$ is closed.
0
votes
1answer
431 views

Locally Bounded vs Bounded Almost Everywhere

Is a locally bounded function $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ also "bounded almost everywhere"? Is the viceversa true? Notes. Definition of "local boundedness": $f: \mathbb{R}^n ...
0
votes
1answer
244 views

Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
1
vote
1answer
1k views

Clarification on bounded convergence theorem.

For the proof of Bounded Convergence Theorem, I see how to get most all the information, but I don't see exactly why $f$ is measurable. I assume I am missing something completely obvious. Here is the ...
3
votes
1answer
202 views

A way to split this integral/norm

My last question hasn't got any replies so I'll try another.. Is there a way to split the following integral ($g$ is arbitrary) $$\int{f^2g}$$ so that I instead have an expression involving the $L^2$ ...
3
votes
1answer
357 views

continuity of power series

I want to prove that every power series is continuous but I am stuck at one point. Let $\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ a power series with a radius of convergence $r>0$ and let ...
2
votes
1answer
98 views

radius of convergence of $\sum_{n=0}^{\infty} (2n+1)(2x)^{2n}$

Determine the radius of convergence of the following power series $\sum_{n=0}^{\infty} (2n+1)(2x)^{2n}$. Is the following correct? $\sum_{n=0}^{\infty} (2n+1)(2x)^{2n} = \sum_{n=0}^{\infty} ...