Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,291
questions
-1
votes
1
answer
80
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Closed on generic set implies closed set whole set [closed]
Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
- 1,011
-1
votes
3
answers
447
views
on compact support distributions [closed]
If $f$ a distribution with compact support then they exist $m$ and measures $f_\beta$,$|\beta|\leq m$ such that
$$f=\sum_{|\beta|\leq m}\frac{\partial^\beta f_\beta}{\partial x^\beta}$$
how to ...
- 57
1
vote
0
answers
92
views
Is harmonic mean of linear functions a Bernstein function?
According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:
$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$
is a Bernstein function, ...
- 93
1
vote
1
answer
96
views
Degree of continous function, a question about its representation
Let $f \in C(\mathbb R,\mathbb R)$, $\text{degree}(f)=\sup\limits_{a \in\mathbb R} \{ \text{card}(f^{-1}(\{a\})) \}$
Is it true that $\forall f \in C(\mathbb R,\mathbb R),\text{degree}(f)=k\in\mathbb ...
- 3,795
3
votes
2
answers
121
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Comparing the tails of two related convergent series
Let $b_1,b_2,\dots$ be positive real numbers such that
$$s_1<\infty\quad\text{and}\quad z_1<\infty,
$$
where
$$s_k:=\sum_{j=k}^\infty b_j\quad\text{and}\quad z_k:=\sum_{j=k}^\infty\frac{b_j}{\...
- 117k
2
votes
2
answers
254
views
Do we have a name for this space?
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Consider the class
$$
\mathcal{F}=\{f\in L^{1}(\Omega):\exists C>0 \text{ s.t. } \int_{U}|f|\leq C\sqrt{|U|},\text{ for any }U\subset \Omega.\...
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0
votes
0
answers
45
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A set of zero harmonic measure 2
Let 𝑉 be a bounded open set in $\mathbb{R}^{m}$, $m\geq 2$, and 𝑊 the interior of the closure of 𝑉. Let 𝐸 be a subset of ∂𝑉∩𝑊 (∂ means boundary) such that:
1) $E$ has positive ($m-$dimensional) ...
- 411
0
votes
1
answer
238
views
Upper bound for sum of $k2^k$
I'm looking at the sum
\begin{align*}
C_p \gamma \sum_{k=1}^N k 2^{k\left(\frac{p}{2} -1 \right)} + 2 n^{\frac{1}{2} - \frac{1}{p}}
\end{align*}
where $C_p$ is some constant depending on $p$ and $\...
0
votes
0
answers
52
views
Reference for inequality for TV of positive measures
Let $\mu,\nu$ be positive measures on some measurable space $(X,\mathcal{F})$.
Let $||\mu-\nu||$ denote the total variation distance between $\mu$ and $\nu$.
Is the inequality
$$ ||\mu-\nu|| \le 2(|\...
- 6,061
2
votes
1
answer
167
views
A set of zero harmonic measure
We know that a set may be of zero harmonic measure without its Lebesgue measure being zero (see Armitage and Gardiner, classical potential theory, pg 178).
Now, consider the following problem. Let $...
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2
votes
0
answers
55
views
Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians
This is a follow-up question of this one.
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...
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1
vote
0
answers
291
views
Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
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0
votes
0
answers
58
views
Gauss lemma for nonsmooth metric
$g_{ij}(x)\in L^\infty(\mathbb{R}^n, M^{n\times n})$ is a metric in $\mathbb{R}^n$ satisfying $\lambda |x|^2\leq g_{ij}x^ix^j\leq \Lambda |x|^2$($\lambda>0$&$\Lambda>0$)
Can we find a ...
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0
votes
1
answer
134
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
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2
votes
0
answers
135
views
Prove the equicontinuity of a maximizing sequence
Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...
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0
votes
0
answers
139
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Derivatives in unusual support domains
Originally posted on Math.StackExchange, here, but I was advised to post it on MathOverflow as it is a research question. Now two final, great answers have been posted, see on Math.StackExchange.
I ...
- 3,088
2
votes
2
answers
272
views
Rational representation of reals [closed]
Can we find for any real number $x$ the sequence of rationals $q_n(x)$ with properties:
$\lim\limits_{n\to\infty} q_n(x)=x$
$q_n(x+y)=q_n(x)+q_n(y)$
$q_n(xy)=q_n(x)q_n(y)$
?
- 1,133
2
votes
2
answers
294
views
Simplify the difference of two dilogarithms--as in the logarithmic counterpart
This question--pertaining to the quantum-information-theoretic topic of "bound entanglement"--stems from the question and answer to https://math.stackexchange.com/questions/3464105/if-possible-...
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2
votes
3
answers
213
views
Equivalence of operators
let $T$ and $S$ be positive definite (thus self-adjoint) operators on a Hilbert space.
I am wondering whether we have equivalence of operators
$$ c(T+S) \le \sqrt{T^2+S^2} \le C(T+S)$$
for some ...
- 23
4
votes
1
answer
90
views
Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant harmonic function (smooth up to the boundary).
Does there exist a sequence of ...
- 6,621
3
votes
1
answer
360
views
Positive part of Cauchy sequence of Sobolev functions is again Cauchy
Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
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0
votes
1
answer
344
views
Mistake in SageMathCell code, finding integral points on elliptic curves [closed]
I've the following number:
$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$
Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
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3
votes
1
answer
195
views
Decomposition of a real analytic variety
Is the following true? If so, I would be grateful for a reference that contains such a result and its proof.
Let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a real analytic function, and $V:=\{\mathbf{...
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5
votes
2
answers
2k
views
Elementary calculus estimate or not?
Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
user69109
8
votes
2
answers
552
views
The limit of a function with derivative at least $1_\mathbb{Q}$
Let $f:\mathbb{R}\to \mathbb{R}$ be differentiable, such that $f'(x) \ge 1_{\mathbb{Q}}(x)$. Is it true that $\lim_{x\to\infty}f(x) = \infty$?
- 89
4
votes
2
answers
540
views
Is there a Borel-measurable function which maps every interval onto $\mathbb R$?
Using AC, one easily defines a function $F:\mathbb R\to \mathbb R$ such that the $F$-image of any real interval $(a,b)$ ($a<b$) is equal to $\mathbb R$.
(Equivalently, the $F$-preimage of any real ...
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14
votes
1
answer
648
views
Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows:
$$S:=\{t\in [0,\infty):nt\...
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1
vote
0
answers
59
views
What is the distance of a particular root to the farthest one with respect to it as a function of a compacting factor?
Let $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined as
$$f := 2a_{1}(x - x^{p}) + 2a_{2}(x - x^{p})\sum_{k \in K}||x_{k} - x^{p}_{k}||^{2} + \dfrac{2a_{3}}{\alpha}\sum_{j \in J}\dfrac{...
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2
votes
2
answers
428
views
If $0 \le \mu(A) < p < 1$, when is it true that there exists a measurable $B \supseteq A$ such that $\mu(B)=p$?
Let $(X,\mu)$ be a probability measure space and $A$ be a measurable subset of $X$ such that $0 \le \mu(A) < p < 1$.
Question
When is it true that there exists a measurable $B \subseteq X$ ...
- 6,726
6
votes
3
answers
581
views
Convex subsets of an open set
Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?
Edit: As Ilya Bogdanov pointed ...
- 157
0
votes
1
answer
134
views
A link between continuity and 0-borelian? [closed]
Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ ...
- 3,795
1
vote
0
answers
85
views
Gaussian width and restricted isometry
It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$.
If a ...
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11
votes
0
answers
319
views
Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
- 3,795
1
vote
1
answer
740
views
Right continuous filtration
In optimal control theory, we often need a filtration do be right continuous. Consider a filtered probability space $(\Omega, \mathcal F, \mathbb P)$ equipped with a right continuous filtration $\...
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4
votes
0
answers
261
views
Dual space of ${\rm Lip}_0(\mathbb R^d)$
This question comes to me when I read this paper : https://arxiv.org/pdf/1702.06049.pdf
Let ${\rm Lip}_0(\mathbb R^d)$ be the space of Lipschitz functions $F$ on $\mathbb R^d$ with $F(0)=0$. Then is $...
user128095
2
votes
2
answers
463
views
Dual space of the completion of the space of Lipschitz functions
This question is a continuation of this post : Metrization of a topological vector space
Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user128095
2
votes
2
answers
327
views
Metrization of a topological vector space
Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ ...
user128095
2
votes
2
answers
470
views
Difference between harmonic mean of arithmetic means and arithmetic mean of harmonic means
Let $S=\{(x_i, y_i)\}_{i=1...n} \in [0,1]^{2n}$ bet a tuple of ordered pairs, and let $A, H$ denote the arithmetic and harmonic mean. Then
$$
\sup_S (H(\underset{i}{A}(x_i),\underset{i}{A}(y_i)) - \...
5
votes
1
answer
144
views
Existence of operator with certain properties
I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated ...
user69109
6
votes
1
answer
464
views
Holomorphic extensions of a non-vanishing real-analytic function
Let f(z) be a holomorphic function defined on an open neighborhood $R$
of the interval $I=[0,1]\subset \mathbb{R}$. Assume $f$
does not vanish on $I$. Then $g(x) = |f(x)|$ is a real-analytic
function ...
- 19.3k
0
votes
0
answers
81
views
equivalent of an alternating series
Let $d_n=\mathrm{lcm}(1,\cdots,n)$. By the prime number theorem $d_n=e^{n+o(n)}$.
I look for an equivalent of the function $\sum_{n\ge0}(-1)^n\frac{d_n}{n!}t^n$ when $t\to+\infty$. Unfortunately, the ...
- 3,737
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 4,969
0
votes
0
answers
80
views
A question about multivariable calculus and optimization
Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$.
Consider the integral :
$$\int_{\bar{...
- 271
6
votes
1
answer
173
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,621
11
votes
3
answers
2k
views
How can I simplify this sum any further?
Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...
- 113
3
votes
2
answers
793
views
Does this version of Clairaut-Schwarz theorem hold when mixed partial derivatives are of order greater than $2$?
I asked this question on MSE here. One person gave an answer but then he deleted it because my version of Clairaut-Schwarz theorem is stronger than his. I meant my version only requires the continuity ...
- 851
2
votes
1
answer
403
views
If $\int_E f = 0$ for all $E$ the translation and dilation of $E_0$ then $f = 0 \text{ } a.e.$
Let $f \in L^1(\mathbb{R}^n)$. It's obvious that if $\int_R f = 0$ for all rectangles $R$ then $f = 0$ $a.e.$ since every open set is union of almost disjoint rectangles and consequently with zero ...
- 1,084
1
vote
1
answer
539
views
Most general form of Jensen's inequality
What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...
- 2,129
1
vote
2
answers
263
views
Is $ \{ \frac{1}{n} + \frac{1}{m} : n,m \in \mathbb{N} \}$ dense in some interval of $\mathbb{R}$? [closed]
Consider the set $\{\frac{1}{n} + \frac{1}{m}: n,m \in \mathbb{N} \}$. Is this set dense in some interval of $\mathbb{R}$?
More generally let $S_k= \{\sum_{i=1}^k \frac{1}{n_i}: n_i \in \mathbb{N} \} ...
- 93
8
votes
2
answers
1k
views
Arzela-Ascoli for L_p-norm
Since I am from a different mathematical field and couldn't find it: Is there something which would be best called an Arzela-Ascoli version for the $L_p$-norm, namely:
Let $X,Y$ be two nice ...
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