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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127 views

Continuous functions with the same Fourier coefficients

I know that if two continuous $2\pi$ -periodic functions $f,g$ have the same Fourier coefficients then $f=g$. Is the assumption about $2\pi$ periodicity of functions essential? Thanks
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2answers
35 views

Sets and length.

I have a set P in R constructed as follows: Let $E_0 = [0,1]=$ {$.d_1d_2... : 0\leq d_j \leq 9 $ for all $j$}. Let $E_1 = ${$x \in E_0 : d_1 \ne 0$} Let $E_2 = ${$x \in E_1 : d_2 \ne 0$} Continue ...
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3answers
74 views

Prove that $n!e-2$ $<$ $\displaystyle \sum_{k=1}^{n}(^{n}\textrm{P}_{k})$ $\leq$ $n!e-1$

Prove that $n!e-2$ $<$ $\sum_{k=1}^{n}(^{n}\textrm{P}_{k})$ $\leq$ $n!e-1$ where $^{n}\textrm{P}_k = n(n-1)\cdots(n-k+1)$ is the number of permutations of $k$ distinct objects from $n$ distinct ...
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1answer
68 views

How can I prove that $3$ is the only solution to the equation $2^n - 2n - 2 =0$ for $n\geq2$?

I'm working on a probability question: Given the equiprobability of "having a boy" and "having a girl" as $1/2$ each, for what value of $n$ births, $n\geq2$ are the following two events independent? ...
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1answer
145 views

Function $k$ times differentiable $+$ root of multiplicity $k$

Problem: Consider the continuous function $f$ which is $k$ times differentiable: $f(\alpha )=f'(\alpha )=\cdots=f^{(k-1)}(\alpha )=0$ and $f^{(k)}(\alpha )\neq 0$. Assume that $\alpha$ is a root to ...
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1answer
264 views

How to show that if $f$ or $g$ is continuous, then the convolution $f \star g$ of those functions is continuous?

How to show that $f \star g$ is continuous if $f$ or $g$ is continuous? Do you use $\epsilon - \delta $ - approach in the proof? some hint. I define $$(f \star g)(x)=\frac{1}{2\pi} \int_{- \pi}^{\pi} ...
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116 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
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2answers
487 views

Riemann-Lebesgue Integrable

Show that there is no riemann integrable function $f$ on $[0,1]$ such that $f=\chi_{C}$ a.e. (almost everywhere), where $C$ is the fat cantor set. Proof: Would it suffice to show that $\chi_C$ is ...
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1answer
73 views

Uniform Convergence Weierstrass

Hi could you help me with the following: If I have a function $g \in C^2(0,R)$ with $|g''(x)| \le M$, i.e. its second derivative is bounded except a finite number of points where at those ...
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1k views

If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<e$

Just having trouble with this problem. First, it says to prove that if $E$ is Lebesgue Measurable, and $e>0$ is arbitrary, then there is an open $O$ such that $E \subset O$ and $m(O\setminus ...
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68 views

Continuity of a compact operator

Consider a separable Hilbert space $(H, \left<\cdot, \cdot\right>)$, and denote with $\mathcal{T}^s$ and $\mathcal{T}^w$ the topology induced by the norm and the weak topology, respectively. Is ...
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1answer
258 views

Why are the integrands dominated by $\alpha f$

This is on page 32 of Rudin's Real and Complex Analysis, 3rd Edition: Suppose $\mu$ is a positive measure on $X$, $f: X \rightarrow [0, \infty]$ is measurable, $\int_X f d\mu = c$, where ...
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1answer
1k views

Closed, bounded interval

Theorem 3.39 of An Introduction to Analysis by W. R. Wade is: Suppose that $I$ is a closed, bounded interval. If $f : I \to \mathbf{R}$ is continuous on $I$, then $f$ is uniformly continuous on ...
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1answer
42 views

Proving the inequality of an integral applied to a sum

For each $n\in\mathbb{N}$ and $x\in\mathbb{R}$ define $$ g_n(x)=2nxe^{-nx^2}\qquad h_n(x)=g_{n+1}(x)-g_n(x) $$ Prove that $$ \int\limits_0^\infty\sum_{n=0}^\infty h_n(t)dt \neq ...
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2answers
340 views

Pattern matching circle, square or triangle

I have a set of x, y co-ordinates that are actually taken from hand drawings of a circle, square or a triangle. Using the set of points, I need to mathematically find if the points approximately fit a ...
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1answer
48 views

Is this proof about metric space correct? (help)

let $M$ a metric space, $a\in M$, $r>0$ and $F = M − B(a, r)$. Show that if $d(x, F ) = 0$, then $x ∈ F $. Proof: let's get a contradiction assuming that $x\notin F$. We have that: ...
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1answer
364 views

continuity of function where the functions restricted to certain closed subspaces are continuous

I was just trying to solve an exercise in Munkres' "Analysis on Manifolds" to get better at continuous functions and topology in $\mathbb{R}^{n}$, and i got stuck. Here it is: Let $X = A \cup B$, $A$ ...
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2answers
266 views

Proof of Goldstine's theorem

On p.98 of these notes, or the first result to come up for the search "Goldstine" one finds a proof of theorem 7.24. http://www.math.uwaterloo.ca/~lwmarcou/Preprints/LinearAnalysis.pdf I don't ...
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1answer
289 views

Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem. If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$. How to prove it?
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1answer
70 views

Extending of domain of smooth function of two variables

Let $f: [a,b]\times [c,d] \rightarrow \mathbb R$ be a smooth function of two variable (assuming that in boundary points $f$ has continuous one side partial derivatives). Is a simple way to extend $f$ ...
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30 views

producing a stock market game based on variable data

I recently launched a stock market game (http://linkdaq.net) based on the top 50,000 sites ordered by the amount of links, it works reasonably well but I'm not 100% happy with the maths behind it. ...
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116 views

I f a compact metric space is compact then any closed unit ball of C(X) will not be compact if X is infinite

Let $(X,d)$ be a compact metric space and let $S=\{f \in C(X): \|f\|=1\}$. Show that if $X$ is an infinite set then $S$ will not be compact. I want to prove it using the concept that if $X$ is ...
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1answer
106 views

Building a graph from pairwise distances

I am not familiar with graphs. However, I am curious about a question on graphs. Given a finite set equipped with a metric, is there any studying on the following problem? Problem: given the metric ...
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2answers
139 views

Back to cauchy sequences.

In this question I will write about a problem which seemingly crops up while proving that all Cauchy sequences are convergent. Consider a Cauchy sequence. As it is a Cauchy sequence it must be a ...
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0answers
23 views

How can one average two semi-partitions of an interval

Suppose that we have an interval $[a,b]$. Define a semi-partition of $[a,b]$ to be any ordered list of points $x_1 < ... < x_n$ all contained in $[a,b]$ ($a$ and $b$ need not be included). ...
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1answer
96 views

Exchanging limits and integrals

Let $\{f_i\}$ be a sequence of pointwise discontinuous functions whose limit is Dirichlet's function. I read that $$\lim_{n\to\infty}\int f_n(x)dx \not= \int \lim_{n\to\infty} f_n(x)dx$$ as the ...
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1answer
53 views

Integrating a power series

I am not a pure mathematician, so I would appreciate some help from people who have done analysis! Can we have some function which is analytic (which I believe just means expressible as a power ...
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1answer
289 views

for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact

Let $(X,d)$ be a compact metric space and let $S= \{f \in C(X):\|f\|\le 1\}$ be the closed unit ball of $C(X)$. Show that if $X$ is an infinite set then $S$ will not be compact.
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Image of smooth manifold is a submanifold

It's know that if $M$ is a compact, smooth manifold of dimension $n$ and the map $f: M \to \mathbb{R^m}$ is injective, smooth, $n \le m$ and $Jf(a)$, the Jacobian, has rank $n$ for every $a \in M$, ...
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18 views

Is it possible to further simplify the following equation?

Is it possible to write the following equation in an even simpler form? (In other words does this have any specific implications on the form $\vec f(\vec x)$ can take?) $${\partial f_j(\vec x)\over ...
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2answers
137 views

Looking for simple function: Passes through 0, sqrt like but never reaches 100

It's in the title. I am looking for simple function that passes through 0, square root like, never reaches 100, but comes closer and closer to it. I'm sure this is very basic, nothing fancy. But I ...
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2answers
52 views

$L^2$-Oscillation

Let $f:[0,1]\to \mathbb{R}$ be a smooth function such that the following property is satisfied. $$\int\limits_{[0,1]}\int\limits_{[0,1]}|f(x)-f(y)|^2dxdy\leq \varepsilon.$$ What can I most say about ...
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3answers
121 views

Power series of a function

I am wondering if there are any functions $f(x)$ such that it cannot be expressed as a power series of $x$? This might turn out to be a silly question, but I can't think of one at the moment. ...
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47 views

Generalization of a problem on canonical transforms of operators

This question is inspired by the solution to Question 1 here. $$H_0={p^2\over 2m}+{1\over 2}mw^2x^2\\$$ Perturbation $$H_1=g{w\over 2}(xp+px),g\in \mathbb R, |g|<1$$ We get rid of the ...
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1answer
163 views

Completing $\Bbb R$ when some “divergent” sequences are Cauchy sequences

If I equip $\mathbb{R}$ with the metric $$ \rho(x,y) := \left|\arctan(x) - \arctan(y)\right| $$ then sequences like for example $x_n = n$ are Cauchy sequences, so it is clear that $\mathbb{R}$ is ...
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1answer
97 views

Is it possible for the value of a residue to be $\infty$?

I'm trying to integrate a complex valued function over the extended real domain $[0,+\infty]$, for which I'm attempting to use the residue calculus. My function has a number of poles above the real ...
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2answers
449 views

Open Set is $G_{\delta}$ set

In any metric space prove that every open set is $G_{\delta}$ set and every closed set is $F_{\sigma}$ set.(Hint: use the continuity of $x\longmapsto d(x,A)$.) I tried to prove this by saying: If $U$ ...
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1answer
159 views

Prove $\ell_1$ is first category in $\ell_2$

Prove that $\ell_1$ is first category in $\ell_2$. I tried to solve this, but had no idea about the approach. Any suggestions are helpful. Thanks in advance.
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1answer
143 views

1-1 functions and Schroder-Bernstein Theorem

Using Schroder-Bernsten Theorem. Assume there exists a 1-1 function $f:X \rightarrow Y$ and another 1-1 function $g:Y \rightarrow X$. If we define $f^{(-1)}(y)=x$, then $f^{(-1)}$ is a 1-1 function ...
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1answer
208 views

Showing the Cantor function is not Lipschitz.

This is one I am having a lot of difficulty with. I'm not sure how to show that the Cantor function (or 'Devil's Staircase) is not Lipschitz.
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119 views

How do I evaluate $k'$ from $\int_{k'}^{\infty} \sqrt{n/2\pi} \, \, e^{(-n/2 \, (\bar x-\mu_0)^2)} \, d\bar x = Z_\alpha$?

$k'$ is supposed to be $\mu_0+ \frac{Z_\alpha}{\sqrt n}$, but I don't know how to get there.
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1answer
707 views

How to prove the strong triangle inequality?

I'm trying to prove that p-adic space is a metric space(also a ultrametric space), but I find it difficult to prove the triangle inequality. So if one can prove the strong triangle inequality, then ...
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1answer
317 views

Differentiation under integral sign (Gamma function)

This might be a silly question, but I'm reading this article about differentiation under the integral sign, and I'm stumped by something that's written early on. The author is giving a derivation of ...
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53 views

which subarea of math text book study about the theory of smooth function?

In another word, which subarea does the theory of smooth function have? I would like to know the list of book on analysis that i could learn more about smooth function.
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1answer
79 views

I need to find a specific maximum principle

I need a maximum principle that says: If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ ...
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3answers
325 views

Level curves on ellipsoid

Let $a,b,c>0$ with $a\leq b\leq c$. Let $E$ be the ellipsoid determined by $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$ Is there a function $f:E\rightarrow \mathbb{R}$ such ...
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1answer
136 views

Help with norm definition

I'd like to show that the following defines a norm on $\mathbb C^n$: $||x||=(a_1^2+a_2^2)^{1/2}$ Where $x=(x_1,..,x_n)$, $a_1$ is the maximum of the $|x_i|$'s and $a_2$ the second maximum. But I ...
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1answer
207 views

Algebraic numbers dense in R

How would I show that the algebraic numbers are dense in R. I know that the rationals and irrationals are dense in R.
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2answers
184 views

Proving algebraic numbers are countable? Simply stated…

Let $n$ a positive number, and let $A_n$ be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$. Using the fact that every polynomial has a finite ...
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1answer
279 views

Reflexivity of a Banach space

I've run into a few problems in which reflexivity of a Banach space is given as a hypothesis. These problems are sometimes of the type where the banach space is specific/concrete, and sometimes it is ...