Skip to main content

Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Filter by
Sorted by
Tagged with
3 votes
1 answer
25 views

How to show that the limit of a sequence is not equal to some value?

In the second chapter of the book Understanding Analysis by Abbott, Example 2.2.6 proved that picking $N>\frac{1}{\varepsilon}$ suffices to prove the convergence of $\frac{n+1}{n}$ to the number $1$...
Nathan's user avatar
  • 31
0 votes
0 answers
23 views

Completely monotone function is analytic

I want to prove the following. Let $I:=[a,b)$ be finite interval, $f$ is completely monotone on $I$. Then it can be extended analytically into the complex z-plane ($z=x+iy$), and the function $f(z)$ ...
vesszabo's user avatar
  • 3,501
-1 votes
0 answers
27 views

How to give this sum a bound?

Let $x,y\in\mathbb{Z},$ consider the sum below $$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$ is there anything I could do to give this sum a ...
Chang's user avatar
  • 329
0 votes
0 answers
10 views

Infinite Summation of Almost Sure Convergent RVs

Suppose we have random variables $X_{n,i}$ for $n\geq1,i=1,\dots,b_n$. Here we suppose $b_n$ is non-decreasing. We know that for each $i$, the sequence $X_{n,i}$ converges to $X_i$ almost sure. Now we ...
Percy Wong's user avatar
2 votes
1 answer
33 views

weak convergence and pointwise implies $L_p$ convergence

Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$. My proof: Since $f^\pm ...
Mr. Proof's user avatar
  • 1,553
0 votes
0 answers
27 views

Continuity of the right shift

In the lecture we consider the right shift$$u:\mathbb{R}\rightarrow X$$ $$ t \mapsto f(\cdot -t)$$ for fixed $f \in X$. If we take for $X = (C^0_b,\Vert \cdot \Vert_\infty)$, hence the space of ...
Noctis's user avatar
  • 265
1 vote
0 answers
31 views

Cauchy-Schwarz in a complex Hilbert Space proof question

my instructor gave me the following proof for the Cauchy-Schwarz inequality in a complex Hilbert Space, but I do not understand the following part: With a fixed value of $|\lambda|$, the left side is ...
mpavlov23's user avatar
  • 123
0 votes
2 answers
74 views

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

When proving using contradiction,can the method specified below be used. lets assume that $\forall a, \exists e > 0; a < e$ and $a > 0$ since the condition holds for all 'a', let $a=2e$ then $...
D.J.D's user avatar
  • 11
0 votes
1 answer
33 views

The radius of convergence using root test

Is the root test sufficient for finding the radius of convergence of any power series ,i.e, does it determine the whole area where a power series converges? if the answer is no can you give an example ...
Haidara's user avatar
  • 45
0 votes
0 answers
15 views

What is this set? Unknown baire generic set

I'm interested in the set given in this question. For clarity, set $$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$ where $\\{q_j\\}$ is ...
JMK's user avatar
  • 815
1 vote
0 answers
20 views

Missing argument in the proof of the Levy-Khintchine representation .

In one proof of the Levy Khintchine representation of the Laplace exponent of subordinators, the following argument is used: "Assume $f_n : [0,\infty) \to [0,\infty)$ is a sequence of non-...
Olivier's user avatar
  • 1,323
0 votes
0 answers
27 views

How to get the following estimate of integral invoving Airy function

$$ \mbox{Define}\quad G(x,y)=\frac{Ci\left(\gamma(x-y_c)\right)Ai\left(\gamma(y-y_c)\right)}{\epsilon\gamma}, $$ where $y>x,$ $y_c$ is a complex number such that $\Re(y_c)>0,$ $\epsilon$ is a ...
Rayyyyy's user avatar
  • 61
-2 votes
0 answers
31 views

Properties of sup and inf for bounded sequences [closed]

Let $(a_n)$ and $(b_n)$ be bounded sequences. How to prove that: (a) $\sup_{k \ge n}(a_k+b_k) \le\sup_{k \ge n}a_k +\sup_{k \ge n}b_k$. (b) $\inf_{k \ge n}(a_k+b_k) \ge\inf_{k \ge n}a_k +\inf_{k \ge n}...
DoRealAnalysis's user avatar
3 votes
2 answers
182 views

Continuity and lebesgue integrability of integral function, proof verification

Suppose that $f\in L^1(\mathbb R, \mathcal B_{\mathbb R} ,m)$ where $m$ is the standard lebesgue measure. For fixed $h$, let us define: $$\phi(x)= \frac{1}{2h} \int_{x-h}^{x+h} f(t) dt $$ Show that $\...
Hyperbolic Cake's user avatar
0 votes
0 answers
45 views

Diophantine approximation and asymptotic for $\dfrac{1}{\sin(n\pi\sqrt{3})}$

I have this exercise and i proved the lemma but i couldn't use it to prove the asymptotic formula i couldn't plug the sin function into the inequality because it change variations maybe some choice of ...
Absaay Ma's user avatar
0 votes
1 answer
15 views

fractional power function inequality

I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result. Let ...
Jabar S. Hassan's user avatar
1 vote
1 answer
27 views

Subset of index that minimizes a sum of real values

Given a series of real numbers $c_1, \ldots, c_n$ with $ n \in \mathbb{N} $, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
mcmat23's user avatar
  • 1,070
0 votes
0 answers
12 views

Converse of bounds on the spectrum of a Toeplitz matrix

The following is from Robert M. Gray's review (https://ee.stanford.edu/~gray/toeplitz.pdf): Lemma 4.1 Let $\tau_{n,k}$ be the eigenvalues of a Toeplitz matrix $T_n(f)$. If $T_n(f)$ is Hermitian, then ...
gen's user avatar
  • 1,518
0 votes
0 answers
29 views

Smallest value of $k$ for which a function approaches $0$ as $x$ goes to $\infty$

I was playing around with factorials on desmos and trying to find some inequality between $x!$ and $\sqrt[x]{x}!$. After a bit I formulated the following question: What is the smallest value of $k$ ...
learner's user avatar
0 votes
0 answers
26 views

Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
Morcus's user avatar
  • 585
1 vote
0 answers
31 views

Taylor's Theorem for functions on the Rational Numbers

I have been looking for different proofs of the Taylor's Theorem with Peano form of the remainder. But all proofs I have found use some form of the Mean Value Theorem or L'Hôpital's Rule (which also ...
Seno's user avatar
  • 163
0 votes
0 answers
63 views

The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]

Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by $$ d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2. $$ Let $\mathbb{R}$ and $...
Saaqib Mahmood's user avatar
0 votes
0 answers
23 views

Fekete's lemma with an extra constant in the index

Say I have an "almost" subadditive sequence $(a_n)_{n\in\mathbb{N}}$ in the sense that $$a_m+a_n\geq a_{m+n+k}$$ for some fixed $k\in\mathbb{N}$. Then do I have that the limit $\lim_{n\to\...
Baguette Boy's user avatar
5 votes
1 answer
159 views

If $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous, then $\frac{\partial f}{\partial x}$ is continuous?

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that both $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous on $\mathbb{R}^2$. Is then $\frac{\partial f}{\partial x}$ also ...
cnikbesku's user avatar
  • 509
0 votes
0 answers
13 views

Integration of the product of a compact supported convolution

I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) ...
Cantor's user avatar
  • 13
0 votes
0 answers
23 views

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

Let $ \hat f : \mathbb Z^d \to \mathbb C $ denote a function in $\ell^p(\mathbb Z^d)$ where $p \in [1,2]$. Let $f : \mathbb T^d = (\mathbb R / 2\pi \mathbb Z)^d \to \mathbb C$ denote the inverse ...
RunningMeatball's user avatar
0 votes
0 answers
44 views

Estimation of Sum of Binomial Coefficients that doesn't Start from Zero

I am having some trouble showing the following estimation: Let $\mu \in (0, 1)$ be an absolute constant. There exists an absolute constant $\eta \in (0, 1)$ small enough such that the number of ...
Partial T's user avatar
  • 583
4 votes
0 answers
56 views

Approximating powers of elements on the unit circle

Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if: $$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| < \...
J. S.'s user avatar
  • 412
9 votes
6 answers
2k views

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

If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
Garrett's user avatar
  • 145
3 votes
0 answers
50 views

An integral inequality involving the Bernoulli polynomials

The classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{z\operatorname{e}^{t z}}{\operatorname{e}^z-1}=\sum_{j=0}^{\infty}B_j(t)\frac{z^j}{j!}, \quad |z|<2\pi. \end{...
qifeng618's user avatar
  • 1,846
-1 votes
1 answer
19 views

Do three points of inequality between convex functions imply inequality over an interval?

Say I have 2 convex functions, $f:\mathbb{R}\rightarrow\mathbb{R}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$. I want to prove that $f(x) > g(x), \forall x \in [l, u]$. I know that $f(l) > g(l)$ ...
Brian's user avatar
  • 1
1 vote
1 answer
42 views

What is the limit of the alternating series $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ as $z\to1$ for a sequence $T_n\sim cn$?

Let $(T_n)_{n>0}$ be an increasing sequence of positive integers and $c>0$ a positive number such that $\lim_{n\to\infty}\frac{T_n}{n}=c$. Write $F(z)=\sum_{n=1}^\infty(-1)^n z^{T_n}$ for a real ...
Mathew's user avatar
  • 33
0 votes
1 answer
45 views

Given a functional equality prove that the function is never $0$

For a function $u:\mathbb{R} \to\mathbb{R}$ with $u(t)^2+u(t)=t^2+t+2$ , $u(0)=1$, Can we prove that the graph of $u$ never touches the $t$ axis ? We can easly see that by adding $0.25$ to both sides ...
Antony Theo.'s user avatar
  • 1,500
-1 votes
1 answer
129 views

A very interesting formula for $\sin x$

Prove that for all $(n,t,x) \in \mathbb{N}^{*}\times \mathbb{R}\times]0,\pi[$ : $\sin(x) = \left( \sum_{k=1}^{n}t^{k-1}\sin kx \right)\left( 1-2t\cos x+t^{2} \right)+t^{n}\left( \sin((n+1)x)-t\sin(nx) ...
Sewshley's user avatar
  • 187
0 votes
0 answers
75 views

Why is a lot of Fourier analysis done on an annulus?

I am studying harmonic analysis from these lecture notes and a lot of results and definitions always assume that the Fourier transform of a function has support in an annulus or a ball. The same ...
CBBAM's user avatar
  • 6,265
0 votes
0 answers
54 views

Fractional regularity and the quotient $\frac{f(x)-f(y)}{|x-y|^\alpha}$

Holder, Sobolev, and Besov Spaces are often used to measure regularity, and in particular, fractional regularity. On one hand, the relation between their respective norms and regularity/...
CBBAM's user avatar
  • 6,265
0 votes
0 answers
53 views

Solution of the "reciprocal of the heat equation"?

I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ ...
Alejandro's user avatar
  • 191
0 votes
0 answers
15 views

Can an unbounded strictly increasing sequence be convergent? [duplicate]

I was doing excercises on convergence of sequences of real analysis but I came up with a problem I don't know how to prove. Note: The book has not shown Cauchy sequences yet and I don't know what they ...
Jery Lazman's user avatar
0 votes
1 answer
57 views

A problem on dirt displacement

Definition. Given a function $f\in L^1(\mathbb{R})$ such that $xf\in L^1(\mathbb{R})$, the quantity $\int_\mathbb{R}xf(x)\,dx$ is called the unnormalized center of mass of $f$ and is denoted $UCM(f)$. ...
aleph2's user avatar
  • 944
1 vote
0 answers
62 views

Rudin's RCA 2.24 theorem : Lusin's theorem

As precised in the title of my question, the context is the book of Walter Rudin : Real and Complex analysis. And especially the proof of theorem 2.24 (Lusin's theorem) which I put below. I have a ...
Laurent Garnier's user avatar
3 votes
1 answer
75 views

Calculus: Is $(x-1)/f(x)$ decreasing?

Suppose $f$ is a continuous function and $f(x) \geq x-1$ for all $x>1$. It is also given that $f(x)$ is a monotonically increasing function in $x>1$. Can I say that $$ \frac{x-1}{f(x)}\quad\text{...
Cantor_Set's user avatar
  • 1,082
1 vote
0 answers
21 views

Doubts about some problems concerning comparability of infinitesimals?

I'm reading Efimov's Higher Mathematics. There is this definition: After that, there are several problems with some proofs, computational exercises, etc. And then, there are the following problems: ...
Red Banana's user avatar
  • 24.2k
1 vote
0 answers
83 views

I don't know how to prove this limit

Let $x=(x_1,\cdots,x_n)$, $\alpha=(\alpha_1,\cdots,\alpha_n)$ $\sum_{i=1}^n\alpha_i=1$, we have $\alpha>0,x_i\ge 0$ for all $i$. Define $$M_t(x,\alpha)=\left(\sum_{i=1}^n\alpha_ix_i^t\right)^\frac ...
Yinuo An's user avatar
  • 370
-1 votes
1 answer
42 views

Prove of rules of operation with limits that diverge

Could someone help to check my proof, I'm not sure if they are rigorous: Given ${a_n}$, ${b_n}$, and ${c_n}$ be sequences of real numbers and let k be a constant, $\lim_{n \to \infty} a_n = \infty$, $\...
DoRealAnalysis's user avatar
0 votes
1 answer
49 views

Legendre transformation is a continuous map

I was reading an article about the Legendre Transformation in Convex Analysis. It was defined like this: Let $\varphi$ be a continuous function on $\left[0,\infty\right)$ and convex on $\mathbb{R}_{&...
MoinsUnPuissanceN's user avatar
0 votes
0 answers
26 views

Function property on balls and scaling

Assume $X$ is a compact metric space and $Y$ is a metric space. Assume $F:X\rightarrow Y$ is a map. Fix $L\geq 1$. Suppose for each $x\in X$, there exists an $\epsilon_x>0$ such that for every $w_1,...
monoidaltransform's user avatar
4 votes
1 answer
90 views

Can anyone explain the constant in this relation

When the following function is plotted in desmos, $$y^x = \frac{(xy)^y}{k},$$ Graph where $k = 1$, i.e. $y^x = (xy)^y$. As k approaches the number $\approx 2.89473713041139$ ($1/k \approx 0....
Nathan Michael's user avatar
1 vote
0 answers
34 views

A lemma of Cielsielski

Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (...
sabrina's user avatar
  • 11
-2 votes
1 answer
46 views

If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$? [duplicate]

Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b] \...
Saaqib Mahmood's user avatar
0 votes
0 answers
50 views

Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]

Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
Saaqib Mahmood's user avatar

1
2 3 4 5
2938