All Questions
Tagged with reference-request real-analysis
419 questions
-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 ...
- 107
0
votes
0
answers
22
views
Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708
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 ...
- 459
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
- 43.2k
4
votes
1
answer
204
views
Stationary phase formula for a complex valued phase
I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form
$$
I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx}
$$
where $\varphi : \mathbb{R} \...
- 2,696
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
0
votes
0
answers
52
views
References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
user537376
3
votes
1
answer
157
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 807
2
votes
0
answers
80
views
Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
6
votes
1
answer
388
views
Decimal expansion definition of real numbers, constructively
The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers.
A real analysis student of mine is working out of the book Real Analysis and Applications ...
- 10.1k
7
votes
1
answer
580
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
5
votes
1
answer
279
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
0
votes
1
answer
106
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 835
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 171
2
votes
0
answers
43
views
Good Polynomial lower estimates for beta function
I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper
Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq ...
- 677
2
votes
1
answer
246
views
Inequality with Hermite polynomials
Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...
- 43.2k
0
votes
0
answers
89
views
Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
-1
votes
1
answer
204
views
Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
1
vote
0
answers
96
views
Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
- 4,359
1
vote
0
answers
92
views
Modulus of Continuity, Heat Flow, and Derivative Estimates
Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by
\begin{align}
(P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right],
\end{align}
where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
- 801
0
votes
0
answers
33
views
Reference request: injectivity of CWT, density of dilations and translations in $L^p$
Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
- 807
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
6
votes
2
answers
755
views
Prove positivity of a binomial sum
Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
- 43.2k
3
votes
1
answer
458
views
Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.2k
1
vote
0
answers
162
views
Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
0
votes
0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
0
votes
1
answer
67
views
Box dimension and graph of Hölder function
In Kamont "ON THE FRACTIONAL ANISOTROPIC WIENER FIELD" (found here : https://www.math.uni.wroc.pl/~pms/files/16.1/Article/16.1.6.pdf), on page 96, it is claimed that,
if a function $f:I^{d}\...
- 73
27
votes
1
answer
2k
views
Is every real number in [0,1] a product of three (or more) Cantor set's numbers?
It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
- 60.5k
21
votes
1
answer
1k
views
Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
- 313
1
vote
1
answer
179
views
The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$
In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
- 677
10
votes
1
answer
571
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 835
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 49
0
votes
0
answers
34
views
It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
3
votes
1
answer
128
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
0
votes
1
answer
297
views
Is there a reference on the space of Lipschitz continuous functions?
I have hard a time finding the specific properties I'm looking for, I'm wondering if there is literature which proves (or disproves) that the space of all Lipschitz continuous functions of some ...
0
votes
1
answer
98
views
Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
- 835
3
votes
4
answers
351
views
What real distributions solve $f'=0$? [closed]
I mean specifically real-valued Schwartz distributions on the real line. That is linear functionals on $C^{\infty}_c(\mathbb{R})$ continuous in the canonical LF topology. My question is, what are ...
- 11.1k
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
5
votes
1
answer
505
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
2
votes
1
answer
127
views
Partition of unity of simplex
Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$
be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an ...
- 1,904