All Questions
5,633 questions
3
votes
1
answer
293
views
Is the minimal modulus of continuity of a $C^k$ function also $C^k$?
Let $f: [0, 1] \to \mathbb R$ be a continuous function. Define the minimal modulus of continuity of $f$, $\omega(f): \mathbb R^+ \to \mathbb R$ by
$$\omega(f)(\delta) := \inf \{\varepsilon > 0 \, \...
- 6,321
0
votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,115
0
votes
0
answers
46
views
Taming families of rate functions
$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$.
Let us say that a family $(r_j)_{j\in J}$ of ...
- 128k
1
vote
0
answers
73
views
Straightening a function supported on a strip
Given a positive smooth function $b:\mathbb{R}^{N+M}\to \mathbb{R}$ which vanishes exactly outisde of some $U\times\mathbb{R}^M$ ($U\subset \mathbb{R}^N$ open), is there another positive smooth ...
- 151
1
vote
1
answer
131
views
Is there a uniform bound on the number of solutions to ${\partial p \over \partial x_i} (x_1,...,x_n) = c_i$ outside a set of measure zero?
Suppose $p(x_1,...,x_n)$ is a polynomial in $n$ real variables whose Hessian is not identically zero. Can we say that there is a set $Z \subset {\mathbb R}^n$ of measure zero and a constant $M$ such ...
- 156
0
votes
0
answers
62
views
Terminology: maps which are bi-Lipschitz on compact subsets
Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
- 5,405
2
votes
1
answer
104
views
Is this function positive on a set of large measure?
Throughout, we denote by $\mu$ the Lebesgue measure on $[0, 1]$.
Let $f \in L^1([0, 1])$ be a nowhere zero function, that is, the set of all $x \in [0, 1]$ such that $f(x) = 0$ is empty.
Suppose there ...
- 6,321
2
votes
1
answer
148
views
Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 463
2
votes
0
answers
71
views
measure corresponding to certain orthogonal polynomials
Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations:
$xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
- 165
4
votes
1
answer
407
views
Lipschitz-regularity of partition of unity
Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
- 195
0
votes
1
answer
516
views
A problem of Fourier transform and Hölder condition
Suppose that $f$ is continuous on $[0,1]$. Thus, $f\in L^1(\mathbb{R})$ and its Fourier transform exists, as
$$ \hat{f}(\xi) := \int_\mathbb{R} e^{-2\pi i x \xi} f(x)dx, $$
which can also be written ...
- 419
4
votes
1
answer
192
views
Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory
In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which ...
- 2,624
9
votes
2
answers
492
views
Rearrangement, conditional convergence, and "placid" permutations
This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
- 20.5k
0
votes
0
answers
76
views
Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
3
votes
2
answers
197
views
Generalization of stationary points to stationary lines (valleys)?
For suitably smooth functions $f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}$, the concept of stationary points, in particular local minima, can as easily be mathematically defined as it is intuitively ...
- 390
2
votes
1
answer
174
views
Gradient of a convex function on $\mathbb{R}^d$, maximum on hypercubes bounded by values in corners?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex.
For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that
$$
(f'...
- 1,095
17
votes
2
answers
2k
views
Explicit and fast error bounds for polynomial approximation
Main Question
This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance.
EDIT (Apr. 23): ...
- 697
1
vote
1
answer
293
views
Expressing the integral over boundary of a domain as an integral over the domain
Let $\Omega \subset \mathbb{R}^2$ be a domain which is "well behaved" (has all "wishable" properties), so as its boundary. For every $u \in C^\infty(\Omega,\mathbb{R})$, I would ...
- 319
3
votes
1
answer
261
views
A kind of vector-valued Littlewood–Paley inequality for arbitrary intervals
The title may be inappropriate, and I apologize for that.
I'm writing a reading report on my harmonic analysis course. My topic is the Littlewood-Paley inequality for arbitrary intervals, which was ...
- 517
0
votes
0
answers
67
views
Asymptotic behavior of the square Generalized Laguerre polynomial
The asymptotic begavior of the Generalized Laguerre polynomial is given in the Book " Formulas and theorems in the special functions of mathematical physics. Berlin: Springer-Verlag; 1966" ...
1
vote
0
answers
144
views
Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
- 131
2
votes
2
answers
1k
views
Fundamental theorem of calculus for Lebesgue–Stieltjes integrals?
Note: Throughout, we denote by $\mathcal L$ the Lebesgue measure on $\mathbb R$.
Let $g: [0, 1] \to \mathbb R$ be a continuous function of bounded variation. Denote by $\mu_g$ its associated Lebesgue–...
- 6,321
1
vote
1
answer
183
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
- 307
5
votes
1
answer
484
views
Equidifferentiable functions
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of continuously differentiable functions. We say that the sequence $f_n$ is equidifferentiable if for every $x \in [0, 1]$ and every $\varepsilon > 0$, ...
- 6,321
3
votes
1
answer
342
views
Is every sequence of functions with uniformly bounded variation almost equicontinuous?
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of functions.
Given a measurable subset $E$ of $[0, 1]$, we say that the sequence $f_n$ is equicontinuous on $E$ if for every $x \in E$, and $\varepsilon &...
- 6,321
7
votes
3
answers
662
views
Asymptotics for $\int\exp( -x t / \log t)dt$
What is the asymptotic growth rate of $$f(x) = \int_e^\infty e^{ - x t / \log t} dt$$ as $x \to 0$?
As an example of what is meant by "growth rate" consider $$g(x) = \int_e^\infty e^{-x t} ...
- 457
2
votes
1
answer
231
views
Inequality with slowly varying functions
Note: I am reposting this question from Math Stack Exchange, which failed to receive an answer after several weeks and a bounty. Also, I believe it fits the requirements for this website, as it ...
- 127
3
votes
1
answer
153
views
Decomposition of rectifiable curves in $\mathbb R^2$
Let $\gamma:[0,1]\to \mathbb{R}^2$ be a rectifiable curve and $\Gamma=\gamma[0,1]$ be its image.
Is it possible to cover $\Gamma$ by a countable collection of sets $N,R_1,R_2,\dots$ such that $N$ has ...
- 14.3k
2
votes
2
answers
231
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 463
1
vote
1
answer
466
views
Zeros of entire functions
Let $f_w:\mathbb C \to \mathbb C$ be an entire function such that $(0,1) \ni w \mapsto f_w$ is real-analytic.
Assuming that there is a dense subset $D \subset (0,1)$ such that for $w \in D$ the ...
- 192
1
vote
1
answer
473
views
Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
- 185
4
votes
1
answer
206
views
Upper right Dini derivative and indefinite integral
I don't understand how the proof of the theorem below works. (Theorem 13.26, Real Analysis, J. Yeh, 2nd ed.)
Let $f$ be a real-valued continuous function on $[a,b]$ such that $f'$ exists almost ...
- 41
7
votes
1
answer
268
views
A differential equation governing compositional inversion
Looking for references for the following theorem.
Given the formal Taylor series/exponential generating function
$$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$
for which the indeterminates $a_n$ and ...
- 10.5k
0
votes
1
answer
137
views
Zeros of entire functions with parameter
Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
- 192
6
votes
1
answer
376
views
Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?
I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
user329770
1
vote
0
answers
202
views
Function uniquely determined by its values at integer arguments
A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
- 3,098
3
votes
1
answer
110
views
Question on the existence/uniqueness of the fixed point
Let $E$ a Banach space ($E$ is the space of continuous functions on $[0,T]$ for my case). Let $F, G: E\times E\to E$ be contraction maps of contraction constant $\epsilon>0$. Given $b\in\mathbb R$, ...
- 1,334
1
vote
0
answers
46
views
Help with a surface of delay differential equations
This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
- 388
16
votes
1
answer
3k
views
Did Euler know (unconsciously) to integrate by differentiating?
Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based ...
- 4,014
2
votes
0
answers
87
views
Maximal function to high power
Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
- 363
0
votes
0
answers
115
views
Parseval identity extension?
I have stumbled upon the following three-dimensional series:
$$\Lambda_p = \sum_{\underline{n}} \left(\frac{\left|n_1\right|}{\left|\left|\underline{n}\right|\right|_2}\right)^p \left|\hat{f}(\...
- 21
0
votes
1
answer
166
views
Construction of holomorphic function
I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that
$|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$.
I will be happy if someone can give me an idea how to do that. I would like also ...
- 35
0
votes
1
answer
83
views
Functional relationship between two quantities
Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by
$$
\alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
- 6,853
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
0
votes
0
answers
80
views
Alternative to the Sampling Theorem / Invertible transform with sampling criteria
I seek a transform $T$ that operates on real-valued $x(t)$, that
Is perfectly invertible
Has discrete counterpart with continuous reconstructor
Provides conditional reconstruction guarantees
...
2
votes
1
answer
136
views
Expressing a vector valued function in terms of its derivatives
Consider a function
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}^m
$$
given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$.
Does there ...
- 23
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
5
votes
1
answer
630
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
3
votes
1
answer
88
views
More on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge0$
A previous question was as follows:
Assume that $f\colon[0,1]\to[0,1]$ is a diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ ...
- 128k