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Mean of periodic function

The statement is simply not true in general. Take $b(t) = \vert \sin(t)\vert$. $b$ is obviously periodic and even continuous and $$2\pi b_0 = \int_0^{2\pi}b(t) dt = \int_0^{2\pi} \vert \sin(t) \vert ...
Noctis's user avatar
  • 344
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Partitioning a sequence of real numbers to a finite number of convergent subsequences

By B-W, a bounded sequence of real numbers always has at least one limit point. Such a sequence (i.e. a bounded sequence of real numbers) could have a finite number of limit points (for example, it ...
Adam Rubinson's user avatar
3 votes

$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $

We have $\sin(\pi-x)=\sin x.$ Thus $$I_n=2\int\limits_0^{\pi/2}e^{-n\sin x}\,dx$$ As $\sin x\ge {2\over \pi} x$ we get $$I_n\le 2\int\limits_0^{\pi/2}e^{-(2n/\pi)x}\,dx={\pi\over n}\left [1-e^{-n}\...
Ryszard Szwarc's user avatar
3 votes

Twice differentiable Lipschitz functions have Lipschitz gradient

Consider $f(x) = \int_{0}^{x}\sin(t^2)\,dt$. Note $f'(x) = \sin(x^2)$ is bounded, while $f''(x) = 2\cos(x^2)x$ is unbounded, so $f$ is Lipschitz and $f'$ is not Lipschitz.
Kakashi's user avatar
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0 votes

Reference Request: Hausdorff–Young inequality for the inverse Fourier seires

The inequality (actually the equality) holds for $p=2$ by the Parseval identity. For $p=1$ the inequality is obvious as if $\widehat{f}\in \ell^1$ then the series $$\sum_{n\in\mathbb{Z}^d}\widehat{f}(...
Ryszard Szwarc's user avatar
2 votes

Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$

Alternative approach: This approach presumes that the problem composer's intent is that $~\epsilon~$ and $~\delta~$ be wrestled with, rather than constructing an argument based on continuity of ...
user2661923's user avatar
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-1 votes

Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$

To complement Robert Z's answer, here is a way that makes calculating the limit easier (albeit, it does not use the $\epsilon-\delta$ definition of the limit directly.) Note that the functions $x^{3}+...
roblich mandervach's user avatar
3 votes

Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$

You should manipulate the RHS in order to find something which goes to zero as $(x,y)\to (-1,2)$. For instance, starting from your work, we have \begin{align*} \left\lvert\frac{x^3+y^3}{x^2+y^2}-\frac{...
Robert Z's user avatar
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0 votes

$z = xy$ intersects with $y = 2x^2$ on a curve. Points on curve are connected to $(0,0,0)$ by a line segment. Area of segment. Solution verification.

The easiest way to parametrize the given curve is to choose $x$ as parameter, because then we have $(x , y, z) = (x, 2x^{2}, 2x^{3})$. Other choices are possible but would only complicate matters. We ...
M. Wind's user avatar
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2 votes
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If a sequence $a_n$ satisfies the following two properties, does $\sum_{k=1}^{\infty} (\sum_{n=1}^{\infty} \frac{1}{a_n^k} - L)$ converge?

I think* the properties: $a_n$ is a positive, increasing sequence $S_k :=\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n^k}$ converges for all $k \in \mathbb{N}$, $\displaystyle\lim_{k \to \infty} ...
Adam Rubinson's user avatar
1 vote
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Lebesgue integral of L^1 function is differentiable

You only need that $F'(x)$ is extant almost everywhere; in that text, this immediately follows from theorem 1.1 alongside the fact that $F$ is absolutely continuous (this is asserted right before thm ...
daisies's user avatar
  • 1,643
0 votes

Proof of absolutely convergent sums over two indices.

I know this question was formulated about 11 years ago, and by this time you probably already know the answers of what you asked. However, there may be other people (just like me) that have similar ...
Mr. Feynman's user avatar
1 vote
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Approximating powers of elements on the unit circle

Lemma: $R$ is adequate iff $R$ is perfect and $\forall x\in[0,1)\setminus\mathbb{Q}\;\exists (r_n)_{n=1}^{\infty}\in R:\; (r_nx)_{n=1}^\infty$ is uniformly distributed (u.d.) mod 1. Proof: For the ...
Varun Vejalla's user avatar
2 votes
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Prove the unbounded sequence $\left\{{a_n} \right\}$ has a subsequence $\left\{a_{p_n} \right\}$ such that $\frac{1}{a_{p_n}}\rightarrow 0$.

It's easier than this. Since $|a_n|$ is unbounded, given any $k\in\mathbb N$, there must exist $n(k)\in\mathbb N$ such that $|a_{n(k)}| \geq k$. You can choose $n(k)$ increasing in $k$ (why?). ...
Landon Carter's user avatar
1 vote

Saying solution of a PDE is continuous to the boundary

No. Let $\Phi$ be the fundamental solution to Laplace's equation (aka the Newton potential). It has a singularity at $0$. Now if $y \in \partial \Omega$ then the map $u(x) = \Phi(x-y)$ satisfies $\...
Glitch's user avatar
  • 8,496
0 votes

$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.

We can formalize the idea of symmetry by using a different change of coordinates: \begin{align} x &= u, \\ y &= v, \\ z &= -w. \end{align} The Jacobian matrix of transformation $\varphi(u,...
Ennar's user avatar
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2 votes
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M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.

$x = t\cos \alpha\\ y = t\sin \alpha\\ z = t(1-\cos \alpha)$ Your issue seems to be with the calculation of the Jacobian. We want to find $\|dS\| = \|(\frac {\partial x}{\partial t},\frac {\partial y}{...
user317176's user avatar
  • 11.5k
0 votes
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$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.

I have lifted your original post and made a few edits to correct errors. $$x+y > 0$$ $$x^2 - 2x + y^2 - 2y + z^2 < 0 \iff (x-1)^2 + (y-1)^2 + z^2 < 2$$ We have a sphere with central point at $...
user317176's user avatar
  • 11.5k
0 votes

Approximating powers of elements on the unit circle

This is clearly a difficult question which will likely take at least some time to be resolved. In this answer, I will document any progress I make towards a solution. Lemma: If $R$ is perfect and ...
J. S.'s user avatar
  • 414
0 votes

Converse of bounds on the spectrum of a Toeplitz matrix

If every eigenvalue of $T_n(f)$, for every $n$, is non-negative, then yes, $f$ is almost surely non-negative. Let us check it by contradiction. Consider (4.14) in Gray’s review: $$ x^* T_n(f) x = \...
Thomas Lehéricy's user avatar
1 vote
Accepted

How can I show that $\bigcap\bigcup_{k=0}^{n-1} [{k \over n}, {k+1 \over n}]^2=\{(x,x):x\in [0,1]\}$

I use $\langle a,b\rangle$; $a,b \in \mathbb{R}$; to denote the point in $\mathbb{R}^2$ with first coordinate $a$, second coordinate $b$, I do this to not overload the notation $(a,b)$, which may be ...
Mike's user avatar
  • 21.2k
1 vote

Discretized Distributions on Rationals?

I would say that "nice", "benchmark" continuous disributions tend to have regular supports - closure of the set on which its density (w.r.t. the Lebesgue measure) is positive. E.g. ...
SBF's user avatar
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1 vote

Discretized Distributions on Rationals?

Expanded from comments: I do not see how you plan to sum (as opposed to integrate) a density (as opposed to a probability mass function) and get 1. You could create an discrete distribution on the ...
Henry's user avatar
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1 vote
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Does this slight modification alter the limit of this sequence?

Taking a logarithm and applying l'Hopital gives \begin{align} y &=\lim_{n\to\infty} \left[1- \frac{x^2}{n}\left(1+\frac{2}{3}\frac{x}{\sqrt{3n}}\right)\right]^{n/2} \\ \ln y &= \ln\lim_{n\to\...
jgd1729's user avatar
  • 485
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How to give this sum a bound?

Expanding on the idea from my comment and the attempt by the OP, I will establish $8\times 15 \times \left(\frac{\pi^4}{90}\right)^2$ is an upper bound for the sum, assuming $x,y\neq0$ in the sum. ...
Steen82's user avatar
  • 808
4 votes
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Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $

Since $\pi(N)\sim N/\ln N$ by the prime number theorem, $$ f(x) = \lim_{N\rightarrow\infty}\frac{\pi(Nx)}{\pi(N)} = \lim_{N\rightarrow\infty}\frac{Nx/\ln(Nx)}{N/\ln N} = \lim_{N\rightarrow\infty}\frac{...
Einar Rødland's user avatar
0 votes

Finding $\lim \frac{(2n^{\frac 1n}-1)^n}{n^2}$.

Here I want to present another solution to this old problem. First observe that $$a_n= \frac{(2n^{\frac 1n}-1)^n}{n^2}=\Big(\frac{2n^{1/n}-1}{n^{2/n}} \Big)^n=\Big(\frac{-(1-2n^{1/n}+n^{2/n})+n^{2/n}}{...
Boris PerezPrado's user avatar
0 votes

Real Analysis Question about Limit points and ε-neighborhoods

To prove the statement If every $\varepsilon$-neighborhood of $x$ intersects $A$ at some point other than $x$, then $x$ is a limit point of $A$. where your definition has $x$ is a limit point of $A$...
K. Jiang's user avatar
  • 8,689
1 vote

Real Analysis Question about Limit points and ε-neighborhoods

It is better to simply construct a sequence converging to $x$. For each $n = 1, 2, \dotsc$, take a point $a_n \in B_{1/n}(x) \cap (A \setminus \{x\})$. Then you have constructed a sequence $a_n \in A \...
André Caldas's user avatar
1 vote

Integral of Thomae's function

Note that for every partition $P$ of $[0,1]$, $L(f,P)=0$. Thus,$\int_0^1f=L(f)$=sup$\{L(f,P):P\; \text{is partition of }[0,1]\}$=sup$\{0\}=0$
suraj tidke's user avatar
0 votes

Understanding Rudin's PMA Theorem 9.17

The key is to understand the notations here. Each quotient in this sum without $u_i$(i.e. $\frac{f_i(x + te_j)-f_i(x)}{t}$) is the ith coordinate in $R^m$ with respect to $u_i$. So basically, (29) is ...
Kai's user avatar
  • 1
0 votes

A corollary of Rolle's theorem

Since you want to base this on Rolle's theorem: From $f'(a)>0$ then for sufficently small positive $h$, we have ($a<a+h<b$ and) from the limit definition of $f'(a)$, $\frac{f(a+h)-f(a)}{h}>...
Hagen von Eitzen's user avatar
2 votes
Accepted

A corollary of Rolle's theorem

This is a specific case of the Darboux theorem. In your case, this is easier, since you are looking for $c$ such that $f'(c)=0$. Since $f$ is differentiable on $[a,b]$, then $f$ is continuous on $[a,b]...
Martigan's user avatar
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2 votes

Nonstandard Analysis research project ideas

There are two approaches to non-standard analysis: (1) the "extension" approach and (2) the axiomatic approach. In the "extension" approach, the real numbers are extended to the ...
Mikhail Katz's user avatar
  • 43.8k
3 votes
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Missing argument in the proof of the Levy-Khintchine representation .

Hopefully there is a more concise answer. I will prove much more than required, namely that $f_n(x) dx$ converges in distribution towards $\delta_0(dx) + f(x) dx$ with $f$ decreasing, and that $f_n \...
Thomas Lehéricy's user avatar
1 vote
Accepted

Proving existence of certain step functions implies integrability

One way to circumvent this problem would be to take the $s_1 = s$ and $s_2 = t$ (you went from one notation to another, that is something to look out for) corresponding to $\varepsilon/2$ instead of $\...
Bruno B's user avatar
  • 5,849
0 votes

Newton approximation in Tao Analysis 1

$$ |f(x) - (f(x_0) + L(x - x_0))| = |f(x) - (f(x_0) + f(x) - f(x_0))| $$ Why would this be the case? Just because we have $$ L = \lim_{\substack{x \to x_0 \\ x \in X \\ x \ne x_0}} \frac{f(x) - f(...
PrincessEev's user avatar
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1 vote
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$A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2 < 4z\}$ limit: $\lim_{n \to \infty}\frac{1}{n} \int_A \frac{y^2z}{ln(x^2+2y^2+n) - ln(n)} \ d\lambda_3$

Sketch: On $A$, $x^2+2y^2<4z-z^2$. The LHS is non-negative, so the RHS must be non-negative which gives $z \in [0,4]$. Thus, \begin{align*} A &\subset B := \{(x,y,z): z \in [0,4], x^2+2y^2<\...
Sounak's user avatar
  • 91
4 votes
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Translates of a set of positive Lebesgue measure cover $\mathbb{R}$?

Let $E$ be fat Cantor set (i.e. a Cantor like set of positive measure). Then $E$ has no interior. If $\mathbb R=\bigcup_n (E+x_n)$ then Baire Category Theorem implies that $E+x_n$ has an intetior ...
geetha290krm's user avatar
  • 39.2k
0 votes

Infinite Summation of Almost Sure Convergent RVs

Note that by setting $X_{n,i} = 0$ for $i>b_n$ the statement is equivalent to $\sum_{i=1}^{\infty} X_{n,i} \to \sum_{i=1}^\infty X_i$ almost-surely as $n\to \infty$. In turn, by considering the (...
raj's user avatar
  • 321
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How to show that the limit of a sequence is not equal to some value?

I will clarify my comment. By definition, $l$ is the limit of the sequence $a_n$ if for every $\varepsilon>0$ you can find $N$ such that $\left|a_n-l\right|<\varepsilon$ for all $n>N$. Thus, ...
Davide Masi's user avatar
2 votes
Accepted

weak convergence and pointwise implies $L_p$ convergence

In general, the statement in the title of the OP is false. In $((0,\infty),\mathscr{B}(\mathbb{R}),m)$, $m$ is Lebesgue measure, define $f_n(x)=\frac1{x-n}\mathbb{1}_{(1,\infty)}(x-n)$. $f_n\...
Mittens's user avatar
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1 vote
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Approximation of a class of measurable functions by simple functions with "compact domain"

This depends on precisely what you meant by "approximated by". If you meant the exact same thing as in the first paragraph, i.e., there exists a sequence $S_n = \sum_{i = 1}^{k_n} c_{(n, i)}...
David Gao's user avatar
  • 9,953
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The radius of convergence using root test

The root test determines the radius of convergence. For the series $\sum_{k=0}^\infty a_k z^k$, we have radius of convergence $R$, where $$ \frac{1}{R} = \limsup_{k \to \infty} |a_k|^{1/k} . $$ This ...
GEdgar's user avatar
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How do I Show that $\exists a, \forall e > 0; a < e \Rightarrow a \le 0$

What you want to show is: if a constant $a$ is smaller than any arbitrary positive number, then it must be non-positive, that is, $a \leq 0$. The problem is that the statement in logical notation in ...
Lucas Cândido's user avatar
2 votes

Constructing the interval [0, 1) via inverse powers of 2

There are already good answers, but just to show that your intuition could be followed to the end. This answer is not completely rigorous and it's certainly more work than just the simple ...
JiK's user avatar
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2 votes

How do I Show that $\exists a, \forall e > 0; a < e \Rightarrow a \le 0$

Be careful when using both existential and universal quantifiers as you can’t swap their order ! For example your first statement is “for all $a$, it exists some $e$ such that blabla”. In this ...
Maxime's user avatar
  • 395
3 votes
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Continuity and lebesgue integrability of integral function, proof verification

This seems to be a correct proof. For the convergence $\chi_{E_n}\rightarrow\chi_E$, consider points $t$ in $\mathbb{R}$ different from from $x+h$ and $x-h$; then there is a positive $\delta$ for ...
Susana Santoyo's user avatar
2 votes

Continuity and lebesgue integrability of integral function, proof verification

I will asume that $h>0$. The case $h<0$ is left to you. $$2h\|\phi_1\|=\int_{-\infty}^{\infty} |\int_{x-h}^{x+h} f(t) dt| dx$$ $$ \le \int_{-\infty}^{\infty} \int_{t-h}^{t+h} |f(t)| dx dt$$ $$=\...
geetha290krm's user avatar
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4 votes

Constructing the interval [0, 1) via inverse powers of 2

You made an interesting query in the comments: $ \def\lfrac#1#2{{\large\frac{#1}{#2}}} $ I was thinking of something like $x = \lfrac{a_1}{2}+\lfrac{b_1}{3}+\lfrac{c_1}{5}+\cdots+\lfrac{a_n}{2^n}+\...
user21820's user avatar
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