Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,269
questions
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Any viscosity solution must be the distance function?
Suppose $U \subseteq \mathbb{R}^d$ is open and bounded. Is it possible anybody could supply a simple proof that any viscosity solution of$$\begin{cases} |Du| = 1 & \text{in }U \\ u = 0 & \text{...
5
votes
1
answer
467
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Covering measure one sets by closed null sets
(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.)
For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval
$[0,1]$, define
$$\newcommand{\card}[1]{\...
1
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0
answers
63
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Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds
Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?
If $V \subseteq U$ is ...
3
votes
1
answer
235
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Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following
Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^...
5
votes
2
answers
328
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a modification on an infinite Bernoulli convolution
The distribution $\nu_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta_{-\lambda^n}+\delta_{\lambda^n})$. This problem has been studied ...
2
votes
1
answer
233
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Automorphism on the unit interval compatible with a measure preserving set function
Cross-posting from math stack-exchange since it's not getting any visibility there.
I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \...
1
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1
answer
340
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Integral kernel smooth
Assume that $Tf(x):=\int_{\mathbb{R}^n} K(x,y)f(y) dy$ is an operator such that $T \in L( H^{-k}, H^k)$ is continuous for any $k$, where $H^k$ is the $k-th$ order Sobolev space on $\mathbb{R}^n$.
...
3
votes
3
answers
231
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sequencial shift on families =flipped powers. How?
Consider the following family of functions
$$f_n(w):=\sum_{k=0}^{\infty}\frac{(-1)^{k-1}}{k!}(k+n)^{k-1}w^k.$$
QUESTION 1. Does the following hold?
$$f_n(w)=-\frac1{n(f_{-1}(w))^n}.$$
Deeper ...
7
votes
3
answers
368
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Does a certain contractive mapping have a fixed point?
Let $f:X\rightarrow X$ be a contractive mapping of a complete metric space satisfying
$$d(f(x),f(y))\leq\alpha(d(x,y))d(x,y)$$
where $\alpha:\mathbf{R}^+\rightarrow [0,1)$, and $\alpha(t_n)\rightarrow ...
2
votes
0
answers
99
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Roots of a partially holomorphic function
Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...
2
votes
1
answer
476
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Are the partial derivatives of a function increasing in both variables measurable?
Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x_1,y_1)\le f(x_2,y_2)$ whenever
$x_1\le x_2$ and $y_1\le y_2$. It is known that the ...
2
votes
1
answer
373
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Sturm Liouville problems for non-classical orthogonal polynomials
It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$
My ...
5
votes
2
answers
1k
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Derivatives of $C^{\infty}$ non analytic function
Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
-3
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1
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447
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Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
11
votes
1
answer
322
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Partition into sets of positive outer measure
Let $\mu^{\star}$ denote Lebesgue outer measure. Suppose $X \subseteq [0, 1]$ and $\mu^{\star}(X) > 0$. Can we divide $X$ into uncountably many sets $\{X_i : i \in I\}$ such that for every $i \in I$...
8
votes
2
answers
967
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Lebesgue outer measure
Denote the Lebesgue outer measure by $\mu^{\star}$. Is there a subset $X \subseteq [0, 1]$ such that $\mu^{\star}(X) > 0$ and $\mu^{\star} \upharpoonright \mathcal{P}(X)$ is a measure (countably ...
2
votes
1
answer
62
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Decompose a function having antiderivatives into bounded components [closed]
Suppose $f:I\rightarrow\mathbb R$ has antiderivatives on an interval $I\subset\mathbb R$. Then $f$ can be decomposed as $f=g+h$, where both $g,h:I\rightarrow\mathbb R$ have antiderivatives. In ...
3
votes
1
answer
873
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What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
2
votes
1
answer
321
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Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
1
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0
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105
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Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
1
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1
answer
1k
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properties of orderd upper and lower semi continuous functions [closed]
$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.
If $f(x_0) = g(x_0) $ for some point $x_0\in M$,
is it ...
7
votes
1
answer
388
views
Does $(\nabla \times F) \cdot F= (\nabla \times F) \cdot \nabla f$ have a solution?
A grad student asked me this question during office hours, and I couldn't for the life of me come up with a proof or counterexample:
For a given $F:\mathbb{R}^3 \to \mathbb{R}^3$, does $(\nabla \...
2
votes
1
answer
142
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The infinite set of $SBV$ function?
Let $u\in SBV(\Omega)$ where by $SBV$ we denote the special bounded variation function and $\Omega\subset \mathbb R^N$ is open bounded.
Let's identify $u$ by its approximation representative (see ...
4
votes
1
answer
857
views
Derivative is Zero on a dense G_delta set
I have the following question:
I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere.
I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G_\delta$-...
2
votes
1
answer
476
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Inverse of pseudo differential operator
Let $\operatorname{Op}_h(x,D)(a)$ denote the Weyl-quantisation of a symbol $a$. Is there an explicit way to invert this pseudo-differential operator in an asymptotic series? By this I mean, can we ...
0
votes
1
answer
287
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Approximation of a $C^{\infty}_c$ function with tensor products of a constant tensor rank
I asked the following question a few days ago:
Approximation of a $C^{\infty}_c$ function by tensor products
However, I then realised that I actually need a stronger result in my proof.
As in the ...
14
votes
2
answers
784
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Integral of power of binomials equal to sum of power of binomials?
Inspired by this MO question about integrating binomial coefficients and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And ...
1
vote
0
answers
99
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simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
1
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1
answer
318
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Is this a superharmonic function?
Hi everyone: Let $ \Omega $ be a bounded open set of $ \mathbb{R}^{N} $, $ N\geq2 $, and $ F\subset \Omega $ with empty interior. Suppose there exists a superharmonic function $ u $ on $ \Omega\...
2
votes
1
answer
562
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Harmonic measure
Hi everyone: Let $ \omega $ be a bounded open set in $ \mathbb{R}^{q} $, $ q\geq 2 $, and $ E $ a subset of the boundary $ \partial\omega $ that has harmonic measure zero in $ \omega $. Let $ V $ be ...
4
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1
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401
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Approximation of a $C^{\infty}_c$ function by tensor products
Suppose that $f \in C^{\infty}_c ( \mathbb{R}^2 )$, i.e. $f$ is a $C^{\infty}$ function with compact support defined on $\mathbb{R}^2$. The following link
Approximation of smooth compactly supported ...
0
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0
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80
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Comparison of two functions
Given a function $f$ from $R^2$ to $R$ satisfying tha following:
$1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$
$2)$ $f$ is a decreasing function on $x$ and on $y$ and $f$...
1
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0
answers
105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
5
votes
2
answers
490
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Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
1
vote
1
answer
113
views
To what extent do integral moments determine a function?
Suppose that $f$ is a many-times integrable function on $[-1, 1]$. We can consider integral moments of $f$, given by
$$ I_n(f) := \int_{-1}^1 \big( f(x) \big)^n dx.$$
My question is: to what extent do ...
2
votes
1
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221
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Are there many "cusps" in a rectfiable star-shaped set?
Let me first recall the definition of density with respect to a measurable set $E$ as follows:
A point $x \in \mathbb{R}^n$ is a point of density $\alpha$ for $E$ if
$$\lim_{r \rightarrow 0} \frac{...
1
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0
answers
169
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Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
1
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0
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277
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
5
votes
1
answer
226
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Approximation of monotone Sobolev functions
Let $f\in W_{loc}^{1,2}(\mathbb R^2)$ be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of $f$ in a precompact open set are attained at the boundary)....
38
votes
4
answers
3k
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Binomial again, and again
Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, ...
1
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0
answers
99
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Existence of a viscosity solution
Setup
I'm trying to find sufficient conditions for the existence of a viscosity solution to the following PDE,
$$
f(t,s,z) + \partial_sf(t,s,z) \\
+ \sum_{i=1}^{\infty} \left[
\partial_{z_i} f(t,s,z)...
2
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0
answers
182
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Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
1
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1
answer
112
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Reference request: regularity of functionals on the space of probability measures
Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
4
votes
1
answer
2k
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Baire's simple limit theorem "almost everywhere"
The Baire's simple limit theorem states that if the functions $f_n : \mathbb{R} \to \mathbb{R}$ are continuous and converge everywhere to a function $f$ then $f$ has a dense set of continuity points. ...
0
votes
1
answer
89
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Estimating pointwise multiplication conjugated by a Fourier multiplier
I asked this question first on MSE but there was no activity.
Let $m(D)$ be a Fourier multiplier and $f$ a known function. I'm trying to estimate the operator
$$Tu=m^{-1}(D)(f(x)m(D)u)$$
in say $H^s$....
11
votes
1
answer
1k
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The Hölder inequality for fractional order Sobolev seminorm?
This question is post on MSE a week ago. I move it here to draw more attention.
Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define
$$
t(\alpha):=\left(\int_I\int_I \frac{|u(x)-u(y)|^\alpha}{...
3
votes
0
answers
153
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asymptotics of the largest real root
Suppose you have a family of polynomials
$$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$
for $n=0,1,2,\dots$.
Further assumptions:
(1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)...
4
votes
2
answers
451
views
Reference request: concave/convex envelope
I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), ...
4
votes
1
answer
987
views
Product of two non-measurable sets
Let $A\subset\mathbb{R}^p$ and $B\subset\mathbb{R}^q$, it’s not difficult to show that $$m^*(A\times B)\leq m^*(A)\cdot m^*(B)$$, where $m^*()$stands for the outter measure in Lebesgue meaning.
If A ...
8
votes
1
answer
4k
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Lax's proof of the change of variables theorem
I am teaching a course on Multivariable Calculus for Graduate students. I came across this nice article by Lax where a special case of the change of variables theorem is proved:
Theorem. Let $f:\...