Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
3,749
questions
0
votes
0answers
18 views
A problem on rate of decay of fill distance?
Let $X$ be a random variable which takes on values from $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)&...
0
votes
0answers
33 views
Asymptotics for fractional Laplacian
This question is motivated by Asymptotic formula for fractional Laplacian
For the equation
$$
\begin{cases}
\lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\
u^\...
1
vote
1answer
76 views
Convergence of increasing rearrangment
Let $A\subset \mathbb{R}$ be measurable such that there are $a,b\in \mathbb{R}$, $a<b$ fulfilling $[b,\infty)\subset A\subset [a,\infty)$. The right rearrangement of $A^{*}$ of $A$ is defined as $A^...
3
votes
2answers
193 views
Is radial part of a Schwartz class function also in Schwartz class?
Let $f\in\mathcal{S}(\mathbb{R}^n)$, Schwartz class. Consider the function $g$ defined on $[0,\infty)$ by $$g(r)=\int_{S^{n-1}}f(rw)d\mu(w),$$
where $d\mu$ is the normalised surface measure of $S^{n-1}...
1
vote
0answers
71 views
$L^2$ convergence of a particular function
I encounter the following problem when I study harmonic analysis by myself:
Given a function $f \in L^2([0,1])$. Let's fix some irrational number $\omega$. For any $N \in \mathbb{Z}^{+}$, let's define ...
4
votes
0answers
183 views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
0
votes
1answer
110 views
Topological characterization of invertible real matrices [closed]
Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries.
Can you give a short non-constructive proof of the existence of a continuous function $M_n\...
1
vote
0answers
31 views
Monotonically increasing and bounded function is in $BV_{loc}$?
For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e..
I'm ...
2
votes
0answers
73 views
$L^p$ estimate of a multiplier operator
I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies:
$$\sum_{n \in \...
1
vote
1answer
61 views
Decide the order of of an integration involving the $\log$ function
Let $$A_n=\int_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$
I want to discribe the order of $A_n$, by geting a progressive formula or a good lower bound for it. The order ...
2
votes
0answers
21 views
Strict Riesz's rearrangement inequality when function is not nonnegative
The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
0
votes
0answers
60 views
Explicit formula for $ (-\Delta)^s \left( \int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-2s} dz\right) $
For $x \in \Omega \subset \mathbb R^N$, is it possible to compute explicitly
$$
(-\Delta)^s \left( \int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-2s} dz\right)
$$
where $(-\Delta)^s$ is the ...
0
votes
1answer
48 views
Integral bound for square of log derivative
I am currently facing the following problem:
Given a polynomial $f(x) = \sum_{s \in S_f} u_s x^s$, $f(0)\neq 0$, $\lvert S_f \rvert \leq t$ (i.e. $f$ is $t$-sparse) with $u_s$ coming as samples from i....
4
votes
2answers
113 views
Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$
Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...
1
vote
0answers
46 views
Stability of non-differentiability under modification on a small set
Definitions:
Let $\mathcal N$ denote the set of continuous, nowhere differentiable real valued functions on $[0, 1]$.
For $0 < \epsilon < 1$, we say $f \in \mathcal N$ has $\epsilon$-robust ...
7
votes
1answer
144 views
Isoperimetric type inequality in $\mathbb{R}^2$
Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows:
\begin{align*}
\mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
-4
votes
0answers
98 views
Basel problem speed of convergence [closed]
There is a Math.SE post arguing the Basel problem converges subgeometrically.
But what is the exact speed of convergence, or are there any other bounds?
2
votes
1answer
76 views
Generalized Selberg integral
I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
8
votes
1answer
237 views
Malgrange preparation theorem with less regularity
(This question was previously posted on MSE
and I decided to post it here too.)
I am studying the proof of the Malgrange preparation theorem given in the book "Stable mappings and their ...
-2
votes
0answers
18 views
Upper derivative of the modified Bessel function of the first kind and order alpha j_alpha? [closed]
I calculated the upper derivative of the modified Bessel function of the first kind and order alpha j_{\alpha} with respect to the variable with the maple program, but I could not show it for example ...
3
votes
2answers
238 views
Solution to simple non-autonomous ODE
Consider the following ODE with parameters $\alpha,\beta,\gamma \in \mathbb R$
$$f'(t)= \begin{pmatrix} \alpha-\beta t & \gamma t \\ \gamma t & -(\alpha-\beta t) \end{pmatrix} f(t).$$
This ODE ...
2
votes
0answers
41 views
Variation of the sum of absolute values of coefficients for shifted Chebyshev polynomials
Setting
Let $\rho \in ]0,1[$, $\varepsilon\in[0,\rho]$, $k \in \mathbb{N}^*$ and
$$P^\varepsilon_k(X) = \tfrac{T_k\left(\tfrac{2(X+\varepsilon)}{\rho+\varepsilon}-1 \right)}{\left|T_k\left(\tfrac{2(1+...
1
vote
0answers
39 views
Superharmonicity of the distance function
Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
2
votes
0answers
111 views
How does the area affect the integral?
Let $\Omega\subset\mathbb{R}^n$ a open bounded set. For any $r>0$ consider the integral:
$$J_\Omega(r)=\int_{\Omega}\frac{|x^s|dx}{r^c+\sum_{i=1}^m|x^{p_i}|r^{d_i}},$$
where $s,p_i\in\mathbb{N}^n$ ...
1
vote
1answer
119 views
Is there a description of the points of the Cantor set on which the Cantor function is differentiable?
Let $C$ be the usual ternary cantor set, and $f$ the Cantor function, or Devil’s staircase associated to it. We know that $f$ is differentiable a.e., and on every point of the complement $C^c$, the ...
1
vote
1answer
67 views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the maximal Hausdorff dimension $d$ (and corresponding Hausdorff $d$-measure) of ...
4
votes
1answer
198 views
A measurable set that acts as a speedometer
Definitions and some motivation:
Say a car is driving on a straight road. All we know is where it starts, and how much time it spends in certain stretches of the road. With just this much information, ...
6
votes
0answers
75 views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
2
votes
1answer
76 views
Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces
I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise:
For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
4
votes
1answer
467 views
Function whose sets of discontinuities and zeros are the rationals
Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$?
...
-2
votes
0answers
58 views
Extension of Liouville's Criterion for Liouville Numbers to Cantor Series [closed]
Is this a valid way to extend Liouville's criterion for Liouville numbers to Cantor Series?
$$0\le\sum_{k=n+1}^{\infty}\frac{a_k}{b_k!}\le\sum_{k=n+1}^{\infty}\frac{b_k-1}{b_k!}$$
$$a_k\ ,\ b_k\ \in\ \...
6
votes
1answer
321 views
Is there a dense planar rational point set within which the distance of any two points is an irrational number?
i.e. could we find a subset $X\subset \mathbb{Q}^2$ such that $\overline{X}=\mathbb{R}^2$ and that for any $x,y\in X$ the distance $|x-y|$ is an irrational number?
I'm considering the following ...
1
vote
1answer
68 views
Condition for the maximum to be non-increasing
Let $u\in\mathcal{C}^1(\mathbb{R}_+\times[0,1],\mathbb{R})$ such that, for any $t\geq 0$, for all $x_0\in[0,1]$ satisfying $u(t,x_0)=\sup_{x\in[0,1]}u(t,x)$, we have
$$\partial_t u(t,x_0)\leq 0.$$
Is ...
3
votes
1answer
133 views
Real part of tail of logarithm
Given a positive integer $n$, consider $f_n = -\min_{|z|=1} \Re \sum_{i>n} \frac{z^i e^{-i/n}}{i}$. What can be said about the growth of $f_n$? How large can it get?
Taking maximum instead of ...
0
votes
2answers
105 views
A question on minimum principle
Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...
0
votes
0answers
109 views
The square of a measure
Notation: We denote by $\mathcal L$ the usual Lebesgue measure on $[0, 1]$. We denote by $\mathcal P = \{a_0, ..., a_n\}$ a partition of $[0, 1]$ and $\Delta \mathcal P := \max_{0 \leq i \leq n} |a_n -...
2
votes
1answer
131 views
Does the derivative of a BV function with no jump part vanish on level sets?
Let $u: \mathbb R^n \to \mathbb R$ be a $BV$ function with no jump part, i.e., writing $Du = D^a u + D^s u + D^j u$ for the decomposition of $Du$ into absolutely continuous, Cantor, and jump part ...
0
votes
0answers
49 views
Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
6
votes
1answer
130 views
Perron-Frobenius and Markov chains on countable state space
The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer:
Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
1
vote
1answer
99 views
Riesz rearrangement inequality
In the Lieb-Loss's book Analysis, they present the Riesz rearrangement in Section 3, Theorem 3.9 (page 93). Note that the functions $f, g, h,$ are all nonnegative. I want to ask whether the ...
2
votes
0answers
48 views
Tail asymptotics of Durfee square identity
This post is related to the problem Asymptotics of a combinatorial series
According to the Durfee square identity:
$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$
where $(q;q)_k$ is ...
1
vote
0answers
61 views
Can a sequence of absolutely continuous functions be rescaled to be equicontinuous?
Given a function $f: \mathbb R \to \mathbb R$, we say $g: \mathbb R \to \mathbb R$ is a topological rescaling of $f$ if $g = fh$ for some orientation preserving homeomorphism $h$ of $\mathbb R$.
Given ...
7
votes
6answers
690 views
Elementary proof that an open subset of $\Bbb{R}^n$ does not have measure zero?
There is an elementary theory of subsets of $\Bbb{R}^n$ of measure zero, namely one defines the volume of a cube in the obvious way and one says that a subset $A$ has measure zero if given any $\...
9
votes
1answer
933 views
Anti Arzela-Ascoli
Notation: We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence $f_n$ of real valued functions on $[0, 1]
$ are equibounded if $\sup_{n \in \mathbb N}...
3
votes
0answers
53 views
Ekeland's standardness-property inheritable?
Ekeland's inverse function theorem gives weak conditions under which a function $f:E\rightarrow F$ between two graded Fréchet-spaces is locally surjective. The theorem requires
the codomain $F$ to be ...
3
votes
4answers
206 views
Integrals involving fractions of exponentials
I am trying to calculate the average degree of a complex network, which requires me to solve for the following integral:
$$\int \mathrm{d} x \frac{\exp{\left[-x -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\...
0
votes
1answer
110 views
Estimating singular double integral
How can I estimate
$$\int_{(0,1) \setminus B_{\delta}(1/2)} \int_{B_\delta(1/2)} \frac{u(y)v(y)}{|x-y|^{\alpha +1}} \, dy \, dx$$ in terms of a positive power of $\delta$ and suitable norms of $u$ ...
0
votes
1answer
36 views
Fractional Laplacian and support
Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$
where $(-\...
7
votes
4answers
320 views
A geometric mean form of the Hermite-Hadamard inequality, for negative powers
The following inequality appeared in the analysis of a random approximation algorithm:
$$
\int_u^{u+1} x^p\ \mathrm{dx} \leq \sqrt{u^p(u+1)^p}\text{, for } -1\leq p\leq 0, u\geq 1.
$$
This resembles ...
3
votes
2answers
149 views
Smooth functions with zeros of infinite order on a closed set
It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(...
