# Tagged Questions

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).

50 views

### Problem with constructing a smooth function with given properties

I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq |x|$...
38 views

### Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
35 views

### Proving Equalities in Analysis

In measure theory, I saw that while proving some "equalities" - $a=b"$ - (such as measure of any type of an interval is its length, ...), the argument goes as follows: We prove that $a\leq b$ ...
54 views

### Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
94 views

### What is the necessary and sufficient condition of linear dependence of $n$ functions?

If $n$ functions are linear dependent, then the Wronskian determinent is zero, While that the Wronskian determinent is zero cannot imply $n$ functions are linear dependent. So what is the necessary ...
108 views

### Fubini's theorem applied to heaviside step functions

First of all, I should probably mention that I am a physicist, not a mathematician so I sincerely apologize for any lack of rigour in my explanation of my problem. Recently, I have been trying to ...
40 views

### Is set with this property is homeomorphic to Cantor set?

(1) $A$ is nonempty subset of $\mathbb{R}$. (2) For all $x<y \in A$ there is $z \notin A$ such that $x<z<y$. (3) $A$ is perfect. Then is there homeomorphism between $A$ and cantor set? ...
205 views

50 views

### Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
67 views

### Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$? This is true if $f$ is smooth but am ...
80 views

### Is the “difference-integral” of an integrable function integrable?

assume $f$ is Lebesgue integrable (on $\mathbb{R}$) and $h >0$. Is it true that then the double integral $\int_{-\infty}^{\infty} \int_{x}^{x+h} f(u) \, du \, dx$ always exists? Intuitively I have ...
90 views

151 views

### Division algorithm for the natural numbers.

I am trying to prove the following statement from Tao's analysis book. Definition of multiplication $ab++=ab+b$. Definition of addition $(a++)+b=(a+b)++$. Let $n$ be a natural number, and let $q$ ...
19 views

93 views

### How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$?

is there a way to explicitely evaluate this integral for natural numbers $n,m$: $$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ Apparently, if $n$ is odd, this integral is zero due to symmetry.
49 views

40 views

### Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
65 views

### Topological properties of $(0,1)\times \{0\}$

I am having a real hard time solving simple proofs involving open sets. I am confronted with this one: Is $(0,1)\times \{0\}$ open? Is it compact? What is its interior? I know $(0,1)$ is open. ...
Fix $y\in \mathbb R;$ and consider the series: $$\sum_{n\in \mathbb Z}\frac{1}{1+(n-y)^{2}}.$$ My Question is: Can we expect to find some constant $C$; so that, $$\sum_{n\in \mathbb Z} \frac{1}{... 1answer 71 views ### If holomorphic \{f_n\}\to f uniformly on compact subsets of U, then do f_n and f eventually have the same number of zeros? Let U be an open subset of \mathbb{C}. Let \{f_n\} be a sequence of holomorphic functions on U such that f_n\to f uniformly on any compact subset K in U. Suppose f is not constant, ... 2answers 124 views ### For holomorphic functions, if \{f_n\}\to f uniformly on compact sets, then the same is true for the derivatives. Let \Omega be an open subset in \mathbb{C}. Let \{f_n\} be a sequence of holomorphic functions on \Omega such that f_n\to f pointwise and converges uniformly on any compact subset K\... 1answer 44 views ### \sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C for all y\in \mathbb R? Fix y\in \mathbb R and s>1. Consider the series:$$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$My Question is: Can we choose r large enough so that I(y)< C for ... 1answer 26 views ### From the given measure \mu, how to construct another measure \mu^{\ast}; so that d\mu^{\ast}(y)= (1+y^{2})d\mu(y)? Put \mu= \sum_{n\in \mathbb Z}c_{n}\delta_{n}; where \delta_{n} is the unit Dirac mass at n. We note that, \mu is a complex Borel measure on \mathbb R and the total variation of \mu, that ... 1answer 48 views ### \int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty  for some large n? Fix y\in \mathbb R. Define,$$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$My Question is: Can we show that I(y)<\infty for some large n\in \mathbb N ? If yes, what is a value ... 0answers 50 views ### Can we expect, h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s}) for h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R) and s>1/2? We put, M(\mathbb R)= The space of complex bounded Borel measure on \mathbb R [With each complex Borel measure \mu on \mathbb R there is associated a set function |\mu|, the total variation ... 4answers 127 views ### a question how to prove:\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2)) I found a complicated question in my textbook, I can't solve it? How to prove$$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$where x\in(-\pi,\pi). My tried method: I tried ... 1answer 219 views ### Eigenvalues gone wild I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ... 1answer 19 views ### Importance of specifying indexing sets I was going through a rudimentary course in mathematical analysis covering Metric spaces and the book opens up with the idea of open sets. While mentioning the property of open sets it cites that if ... 0answers 26 views ### The variation of Calderon reproducing formula I'm reading the book 'Classical and multilinear harmonic analysis, Muscalu'. I fail to understand the page 261. Actually, I doubt that the proof is right. Let f \in BMO(\mathbb{R^d}) have compact ... 0answers 52 views ### Is Sobolev space H^{s}(\mathbb R), for s>\frac{1}{2}, closed under point wise multiplication? [duplicate] We note that, L^{2}(\mathbb R) is not closed under point wise multiplication. Let s>\frac{1}{2}; and we define Sobolev space, as follows: H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\... 1answer 98 views ### Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces Let's (X_1,d_1), (X_2,d_2) be compact metric spaces such that for every finite subset of X_1 like A (respectively any finite subset of X_2 like B ) there exists a finite subset of X_2 like ... 1answer 86 views ### how to determine the existence of double limit? Let f(x,y) be a function of two variables. Are there any criterions to determine the existence of double limit$$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$If for all y\in(y_0-\delta,y_0+\delta), \... 1answer 71 views ### direction limits and double limit Let f(x,y) be a function of two variables. What is the counterexample that there exists A s.t. for all \theta,$$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$but double limit$$ \lim_{(x,y)\...
Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...