All Questions
77 questions
23
votes
2
answers
2k
views
Which smooth compactly supported functions are convolutions?
If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
- 1,017
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
14
votes
6
answers
6k
views
Russian Equivalent of Big Rudin
Is there any Russian-authored textbook on Analysis equivalent to Big Rudin (Real and Complex Analysis)?
I like Russian math textbooks a lot. I am looking for Russian textbooks (either in English or ...
- 149
11
votes
3
answers
1k
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
- 283
11
votes
1
answer
411
views
A density question for the Hilbert transform
Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions
$$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
- 4,143
11
votes
1
answer
582
views
An extension of the Carlson's theorem in complex analysis
For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
- 4,143
10
votes
1
answer
349
views
On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,143
10
votes
0
answers
845
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
- 2,454
9
votes
1
answer
608
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,143
7
votes
1
answer
370
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 71
7
votes
3
answers
385
views
On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?
Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$.
We say that $f\in C(K)$ is a generator of $C(K)$ when the set $...
- 185
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,143
5
votes
1
answer
353
views
Family of functions with prescribed derivatives
Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
- 4,143
5
votes
2
answers
338
views
Approximation of analytic function by a fixed number of monomials
This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...
4
votes
1
answer
636
views
Existence of a smooth compactly supported function
Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...
- 4,143
4
votes
2
answers
360
views
Functions with asymmetrically decreasing Fourier transform?
$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
- 3,584
4
votes
1
answer
390
views
Existence of periodic solution to ODE
We shall consider the matrix-valued differential operator
$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$
This is ...
- 192
4
votes
1
answer
317
views
Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
- 73
4
votes
1
answer
305
views
Holomorphic extension of the Fourier transform of a measure
If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
- 63
4
votes
0
answers
117
views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
- 521
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
4
votes
0
answers
716
views
Can one integrate around a branch-cut?
How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
- 1,893
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
3
votes
1
answer
425
views
Regularity of boundary of a level set of a $C^{1,\alpha}$ function
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$.
What i want to ask is, if $S_C$ is nonempty for some $...
- 379
3
votes
1
answer
137
views
Estimate the homogeneous components of a polynomial against its maximum
Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over ...
- 463
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 31
3
votes
2
answers
498
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user147204
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
1
answer
237
views
Isoperimetric inequality for analytic functions on an annulus
Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
- 434
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
3
votes
0
answers
235
views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
- 31
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
2
votes
2
answers
281
views
Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
2
votes
1
answer
112
views
On compactly supported functions with prescribed sparse coordinates
Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
- 4,143
2
votes
1
answer
103
views
A density question
Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
- 4,143
2
votes
1
answer
145
views
Estimate for an oscillatory integral of the first kind
I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...
- 159
2
votes
1
answer
163
views
Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)
In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven:
Lemma:
Let $p, q\in \mathbb{R}[x]$ be stable monic ...
- 203
2
votes
0
answers
72
views
Semilinear elliptic equations in complex plane
Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
- 4,143
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
2
votes
0
answers
163
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,143
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
2
votes
0
answers
70
views
The $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+b}{2},\frac{a+b}{2})(z)$ [closed]
The proposition the OP wants to prove is incorrect. --Aug 8, 2017
Can we find a elegant way to prove that the $n$-th derivative for $z$ at $z = 0$ of $F(a,b)(z)$ is no less than that of $F(\frac{a+...
- 1,551
1
vote
3
answers
207
views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
user121558
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,143
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,143
1
vote
2
answers
588
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
1
vote
2
answers
106
views
'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 463