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16 votes
1 answer
969 views

Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis

While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
Tobias Diez's user avatar
  • 5,824
6 votes
3 answers
748 views

Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
Fatima Majeed's user avatar
5 votes
2 answers
355 views

Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?

Is it true that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$ ? Or in other words can you show that the higher order derivatives of the reciprocal of the Riemann zeta function ...
Haidara's user avatar
  • 178
7 votes
1 answer
554 views

Example of continuous function which is not differentiable everywhere in a strong sense

Is there a continuous function $$u\colon (0,1)\to \mathbb{R}$$ such that at every point $x\in (0,1)$ one has $$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$ In particular $u$ is not ...
asv's user avatar
  • 21.8k
15 votes
1 answer
764 views

Does there exist a nowhere smooth function, that has arbitrary many derivatives?

I'm sorry if my title sounds misleading, I don't know a better way to word my question briefly. But I have the following question about functions. First, as long as $A$ is a dense subset of $\mathbb{R}...
Sam Forster's user avatar
12 votes
1 answer
394 views

Is $X\times X$ homeomorphic to $X$ for a space of probability measures?

Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if $$ \int_S ...
Christian Remling's user avatar
2 votes
1 answer
462 views

About the number of critical points of a function

Suppose that $f$ is a totally monotone function on $(0,\infty)$, so that $(-1)^n f^{(n)}\ge0$ for all $n=0,1,\dots$, $f(0+)\in(0,\infty)$, and $f(t)\sim\frac{1}{t^{\frac{3}{2}}}$ as $t\to\infty$. Can ...
Ervand's user avatar
  • 51
1 vote
1 answer
316 views

The number of intersection points of two completely monotonic functions

Is there a way to show that two completely bounded decreasing monotonic function which are also differentiable over $[0,\infty)$ intersect at most at one point? or showing some counterexamples. ...
Ervand's user avatar
  • 51
5 votes
1 answer
206 views

Compactness in trace class operators space

Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$. Are there easy ...
lulli_'s user avatar
  • 59
9 votes
1 answer
292 views

What are the points of the algebra of polynomial functions on an arbitrary vector space?

Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
Dima Roytenberg's user avatar
0 votes
1 answer
239 views

Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]

Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
H Tomasz Grzybowski's user avatar
4 votes
1 answer
274 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
3 votes
1 answer
189 views

Ribe's Theorem: finitely representability between two uniformly homeomorphic Banach spaces

An infinite-dimensional Banach space $X$ is said to be crudely finitely representable (with constant $\lambda$) in an infinite-dimensional Banach space $Y$ if there is a constant $\lambda>1$ such ...
Xiangbo's user avatar
  • 33
2 votes
1 answer
236 views

Self-adjointness of generator and semigroup of an SDE

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
2 votes
1 answer
155 views

Does the support of a smooth compactly supported function have a smooth boundary?

Let $U$ be a bounded domain in $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ an infinitely differentiable function with compact support. My question is whether $\operatorname{supp}(f)$ has a smooth ...
Alexander Wolfram's user avatar
0 votes
1 answer
142 views

Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]

Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
user530909's user avatar
3 votes
1 answer
144 views

Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?

Write $g$ as the inverse of $f$. Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that $$ \displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0 $$ as $r\to0$? If you like,...
Chris Sanders's user avatar
0 votes
1 answer
188 views

Does the second Bourgain–Delbaen space belong to C_p?

The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$. An operator $T:X\to Y$ ...
Ioana Ghenciu's user avatar
0 votes
1 answer
169 views

Existence of a "universal" measure-preserving transformation on the unit interval

Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user avatar
3 votes
1 answer
128 views

Comparing two different principles of premeasure-to-measure extension

It is well-known that a premeasure $\mu_0$ (possibly taking infinite values) on a ring of subsets $\Omega_0$ of a set $X$ can be extended to a complete measure space $(X, \Omega_C, \mu_C)$ ($C$ for ...
Atom's user avatar
  • 133
0 votes
2 answers
148 views

Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines

I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$: $$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$ where $H_m(x)$ is the $m-$th Hermite polynomial....
Darius's user avatar
  • 21
2 votes
0 answers
225 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
-3 votes
1 answer
194 views

Bounding a number-theoretic integral

Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH. My attempt here is ...
charlie_beck's user avatar
1 vote
1 answer
151 views

Is smoothness preserved under an isometric isomorphism?

Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
Tuh's user avatar
  • 113
0 votes
1 answer
157 views

Weak convergence of $f(x,e^{itx})$

This is the desired result (what I want to prove): $$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$ Given that $f\in C([a,b]\times\{e^{i\...
Quý Nhân's user avatar
5 votes
0 answers
204 views

A proof for an $L^p$-$L^p$ inequality

This is a transfer of the question https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
Medo's user avatar
  • 852
0 votes
1 answer
169 views

Is the evolution family self-adjoint?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
Akira's user avatar
  • 835
6 votes
0 answers
212 views

Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck

In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
Cody Morrin's user avatar
0 votes
1 answer
115 views

Fourier transform of exponential over torus

I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
António Borges Santos's user avatar
5 votes
0 answers
163 views

Does this weak omniscience principle have a name?

In constructive analysis, I'm looking at principles which follow both when there exists at least one discontinuous function from $\mathbb{R}$ to $\mathbb{R}$ (equivalent to WLPO i.e. $x > 0$ or $x \...
saolof's user avatar
  • 1,947
1 vote
1 answer
90 views

Sobolev inequality with weight in the case $1<n\leq p$

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
Shaq155's user avatar
  • 459
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
158 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
M.González's user avatar
  • 4,461
0 votes
0 answers
113 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
S-F's user avatar
  • 63
3 votes
1 answer
76 views

How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?

A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
D.R.'s user avatar
  • 831
6 votes
0 answers
131 views

Do there exist strictly contracting eikonal functions on $\mathbb R^n$?

A function $f: \mathbb R^n \to \mathbb R$ is said to be a strict contraction if $$|f(x) - f(y)| < |x - y|$$ for all $x \neq y$. A function $f$ is said to be eikonal if it is differentiable ...
Nate River's user avatar
  • 6,195
1 vote
1 answer
59 views

Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
user2103480's user avatar
1 vote
0 answers
111 views

References on the partial trace

For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows : $$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
Fawen90's user avatar
  • 1,399
-1 votes
0 answers
94 views

Why define Schwartz by supremum rather than limit?

The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite. However,...
Ponder Stibbons's user avatar
1 vote
1 answer
117 views

Lower bound for a commutator trace

I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
Chilperic's user avatar
  • 121
1 vote
1 answer
69 views

Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor

Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
abcdmath's user avatar
  • 105
3 votes
0 answers
154 views

Colimits in commutative Banach algebras?

Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
user577413's user avatar
3 votes
0 answers
91 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
101 views

An inequality related to Problem 10210 AMM 1992 No. 3

Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that $$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
River Li's user avatar
  • 1,053
4 votes
0 answers
90 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 835
3 votes
0 answers
95 views

Deeper reason for why classical orthogonal polynomials have simple generating functions?

Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
Plemath's user avatar
  • 312
2 votes
0 answers
77 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
85 views

Density of a subset of Schwartz space in the fractional Sobolev space

It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
Nirjan Biswas's user avatar
3 votes
2 answers
153 views

On nowhere differentiability of functions that just barely fail to be Lipschitz

By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
Nate River's user avatar
  • 6,195