All Questions
Tagged with real-analysis reference-request
419 questions
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 2,670
-3
votes
1
answer
361
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
- 12.9k
4
votes
2
answers
592
views
From Zurab's integral representation for the Apéry's constant to almost impossible integrals
I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
- 791
1
vote
1
answer
484
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Convexity at a point and Jensen inequality
I am looking for a reference for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". ...
- 128k
4
votes
1
answer
116
views
$AC^p$ curves and pointwise metric speed in abstract metric spaces?
For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s)...
- 1,027
4
votes
1
answer
539
views
Derivatives of Riemann $\xi$ and traces of zeros
Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
- 7,024
1
vote
2
answers
859
views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
- 53.3k
12
votes
2
answers
697
views
Is the square root of a monotonic function whose all derivatives vanish smooth?
Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$.
Is $\sqrt g$ is infinitely (right) differentiable at $x=0$?
...
- 6,741
5
votes
0
answers
163
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Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 39k
1
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1
answer
195
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Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s&...
- 128k
18
votes
4
answers
4k
views
Problems in advanced calculus
I have been teaching Advanced Calculus at the University of Pittsburgh for many years. The course is intended both for advanced undergraduate students and the first year graduate students who have to ...
- 4,784
2
votes
1
answer
71
views
Estimates on divergence-type operator for the matrix
Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
- 52.3k
4
votes
0
answers
125
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Is there a name for this slightly stronger version of Cesàro convergence which "more quickly ignores earlier terms"?
Let $V$ be a normed vector space, let $l \in V$, and let $(a_n)$ be a sequence in $V$. We say that $a_n$ is Cesàro-convergent to $l$ if $\frac{1}{n}\sum_{i=1}^n a_i \to l$ as $n\to\infty$.
Now I will ...
- 708
9
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1
answer
1k
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Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
-2
votes
1
answer
147
views
Asymptotics for certain integrals
I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a ...
- 61.9k
4
votes
1
answer
208
views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
- 16.2k
4
votes
3
answers
499
views
Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
This is a follow-up on the previous question.
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb ...
- 128k
8
votes
1
answer
384
views
Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...
- 17.2k
33
votes
4
answers
2k
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Hahn-Banach theorem with convex majorant
At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
- 28k
2
votes
1
answer
487
views
Difference quotient for functions of bounded variation
Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
- 28k
2
votes
0
answers
200
views
The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
CommunityBot
- 1
0
votes
0
answers
84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
- 109k
1
vote
1
answer
644
views
Most general form of Jensen's inequality
What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...
- 128k
1
vote
1
answer
167
views
Convex-like properties of the polar parametrization of the boundary a convex body on the plane
Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization
$\mathbf p:\mathbb R\to \partial B$ assigning to ...
- 109k
5
votes
2
answers
594
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
- 32.5k
1
vote
1
answer
174
views
mollifier satisfying moment conditions
I wish to find a mollifier $\psi\in C_0^{d+1}(-1,1)$ such that
$$
\int_{-1}^1 x^k \psi(x)dx = \begin{cases}
1, & k=0;\\
0, & k=1,\dots,d.
\end{cases}
$$
This paper (https://home.cscamm....
- 17.2k
2
votes
0
answers
1k
views
bounds on derivatives of mollifiers/mollified functions
Consider the standard mollifier
$$
\phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1.
$$
such that $\int\phi(x) = 1$.
Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am ...
- 640
31
votes
2
answers
3k
views
Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned ...
CommunityBot
- 1
21
votes
1
answer
564
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 9,722
2
votes
1
answer
153
views
Reference for a theorem on subharmonic functions
I need a reference for a theorem that states: Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. Let $u$ be a subharmonic function on $D$. ...
- 17.2k
4
votes
1
answer
1k
views
Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?
It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
- 1,587
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
- 42.8k
1
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0
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511
views
Weak derivative under the integral sign
Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...
- 161
0
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1
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133
views
Product of sets with the Radon-Nikodym Property (RNP)
I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.
Does the above result ...
- 59
1
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1
answer
83
views
Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set
I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
- 29.8k
2
votes
1
answer
139
views
The radius of an interval's image through a space-filling curve
Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
2
votes
1
answer
307
views
Box counting dimension of a set and Lipschitz functions
If $f$ is Lipschitz, then the following holds for the Hausdorff dimension:
$$\dim_H f(A) \le \dim_H A.$$
Is the same true for the box counting dimension?
- 14.2k
2
votes
2
answers
279
views
A conjecture concerning symmetric convex sets [closed]
Question:
Let's suppose that $S \subset \mathbb{R}^n$ is convex and symmetric so:
\begin{equation}
x \in S \iff -x \in S \tag{1}
\end{equation}
Now, if we define the radius of $S$ as $R$ such that:
\...
CommunityBot
- 1
3
votes
1
answer
173
views
Weak Lebesgue spaces and an estimate for BV functions
Let $u \in BV(\Omega \subset \mathbb R^N, \mathbb{R}^N)$. Is it true that there exists a function $f$ in the weak $L^1$ space such that
$$|u(y)-u(x)| \le |x-y|\big|f(y) - f(x)\big|$$
holds for a.e. $...
- 1,277
2
votes
0
answers
269
views
Extending Green's theorem from very special regions to more general regions
Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
- 151
5
votes
1
answer
499
views
Hausdorff dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...
- 839
1
vote
1
answer
247
views
Equivalent notion of approximate differentiability
Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...
- 1,277
-2
votes
1
answer
214
views
About infinite products and Euler Gamma functions [closed]
I am interested in knowing how to calculate infinite products like (or reading any reference about it):
$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$
Inserting it into ...
- 54.2k
0
votes
1
answer
152
views
Reference request: Baire class 2 functions
There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
- 367
11
votes
3
answers
2k
views
Does anyone recognize this inequality?
In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
- 757
3
votes
0
answers
55
views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
- 1,385
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
CommunityBot
- 1
1
vote
2
answers
279
views
Reference request: Functions of bounded variation in one real variable
Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!
- 659
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 28k