# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,703
questions

**3**

votes

**0**answers

43 views

### Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...

**4**

votes

**1**answer

131 views

### Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.
How can I compute the ...

**2**

votes

**0**answers

79 views

### A question about how to use the convexity condition?

At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say that since $p$ is convex, we can deduce that
$$ \...

**1**

vote

**1**answer

97 views

### Hermite Transform of Tanh

I would like to understand $L^2(\mathbb{R},\mu)$ approximation by polynomials of $\tanh$ up to degree $n$ where $\mu$ is the standard Gaussian distribution. This leads to considering the Hermite ...

**4**

votes

**1**answer

112 views

### A continuous bi-Lipschitz shrinking of a domain into a compact subset

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is:
(1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...

**0**

votes

**1**answer

188 views

### Does there exist a continuous nonconstant function $f$ that maps almost all irrationals to rationals? [closed]

Let $f$ be a continuous nonconstant function on the reals. Could it map almost all irrationals to rationals?
This is impossible if $f$ maps all irrationals to rationals, by a well known result.
This ...

**1**

vote

**2**answers

121 views

### Global estimate to an L1 function whose Laplacian is a bounded measure

Pretty simple question:
Suppose that $u \in L^1(\mathbb{R}^N)$ is such that $\Delta u \in \mathcal{M}(\mathbb{R}^N)$ (i.e., $\Delta u$ is a bounded Radon measure). Does $\nabla u \in L^1(\mathbb{R}^N)$...

**35**

votes

**12**answers

5k views

### Examples where existence is harder than evaluation

In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, ...

**1**

vote

**2**answers

91 views

### Non-asymptotic upper bound of right tail of Gamma function

I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?

**43**

votes

**7**answers

4k views

### Two (probably) equal real numbers which are not proved to be equal?

Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal?
I never really understood the assertion that "the reals do not have decidable equality"...

**2**

votes

**0**answers

83 views

### Discrete Sobolev embedding

It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...

**1**

vote

**0**answers

64 views

### Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...

**2**

votes

**1**answer

222 views

### Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$

Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...

**1**

vote

**1**answer

298 views

### Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere

Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
$g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
$n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...

**5**

votes

**2**answers

222 views

### Bounded weak derivative

Let $f \in L^{\infty}$ be a function such that $f$ and the weak derivatives $D^{\alpha}f\in L^{\infty}$ exist for all $\vert \alpha\vert\ge 2$. Does this imply that also $D^{\alpha}f$ with $\vert \...

**1**

vote

**0**answers

48 views

### Dependency of the Wasserstein metric on its parameters

Let the population on some region $\Omega\subset\mathbb R^d$ be modeled by a density function $\rho:\Omega\to (0,+\infty)$. Provided $n\ge 1$ food trucks labeled by their capacity $p_1,\ldots, p_n\in (...

**1**

vote

**1**answer

178 views

### If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set

If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...

**3**

votes

**1**answer

177 views

### Rademacher, maxima, convex hulls

Let $F\subset \mathbb{R}^n$ be a finite set and $\sigma$ be uniformly distributed over $\{-1,1\}^n$. The usual Rademacher average of $F$ (modulo normalizing factors) is
$$ R_n(F)=\mathbb{E}_\sigma \...

**4**

votes

**0**answers

147 views

### Signed variant of the Flint Hills series

I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...

**2**

votes

**1**answer

68 views

### Integral substitution involving the length and angle of two vectors

Let $F\colon\mathbb R^3\to\mathbb R$ be a compactly supported smooth function. I want to compute
$$ \int_{\mathbb R^n}\int_{\mathbb R^n} F(\lVert x\rVert^2,\lVert y\rVert^2,\langle x,y\rangle)~\...

**1**

vote

**1**answer

79 views

### Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$

Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$.
Question
Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-...

**2**

votes

**2**answers

142 views

### Lower bound Renyi divergence between two discrete probability distributions

I am trying to understand the proof of Lemma 1 in this paper (Section 9.2).
The proof shows that given a discrete probability distribution $P=(p_1,p_2,...,p_k)$ where $p_1 \geq p_2 \geq ... \geq p_k$,...

**7**

votes

**1**answer

207 views

### Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...

**6**

votes

**1**answer

352 views

### Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?

Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...

**5**

votes

**0**answers

194 views

### A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$.
Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$.
...

**3**

votes

**0**answers

42 views

### Controlling a Schwartz kernel near the diagonal

Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...

**1**

vote

**2**answers

76 views

### Asymptotic Behaviour of Solutions to a Riccati-type ODE with Small Forcing Term

Consider the following ODE:
$$f'(r) = f^2(r) + O \left( \frac{1}{r^4} \right)$$
as $r$ goes to infinity. The initial conditions are $f(1) = C <0$.
What is the behaviour of a solution $f$ at ...

**0**

votes

**1**answer

64 views

### Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]

Let $f:[0,1] \to \mathbb{R}, G = graph(f)$.
If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$
What technique can I ...

**0**

votes

**0**answers

18 views

### Bounds on the Hausdorff dimension for an IFS satisfying the OSC

Let $([a,b],F)$ be an IFS where $F$ satisfies the open set condition for an open interval $G$. Furthermore, assume that each $f_i \in F$ satisfies: $$\forall x,x' \in \stackrel{-}{G}.q_i |x-x'| \le |...

**2**

votes

**0**answers

67 views

### How we can do the derivative for this equation w.r.t.to time t>0

Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...

**2**

votes

**2**answers

271 views

### Uniqueness of the limit sequence of discrete probability measures

Let $N_n$ be a sequence of natural numbers increasing to infinity, and suppose we have a sequence of finite sets of distinct points
$X_n = \{x_1^{n},x_2^{n},\ldots,x^{n}_{N_n}\} \subset[0,1] \subset \...

**1**

vote

**0**answers

78 views

### An oscillatory integral estimate

Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...

**2**

votes

**1**answer

88 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. $...

**0**

votes

**0**answers

59 views

### Convergence of Schur-Hadamard multipliers

Picking up on the question asked here:
Convergence of sequence of images of Schur multipliers
I was wondering if the same holds in the following setting:
Let $K$ be an integral operator acting on $...

**5**

votes

**2**answers

182 views

### Density character of a metric space is an Ulam number

I am reading this paper and I came across the following sentence:
Throughout the paper we silently assume [...] that the density character (i.e. the minimum cardinality of a
dense subset) of ...

**1**

vote

**1**answer

66 views

### Independent identical distribution sequence

given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...

**1**

vote

**1**answer

101 views

### How to prove the following Whittaker formula

I am a theoretical physicist and
I need help in proving the alternate Whittaker formula
$W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , ...

**1**

vote

**0**answers

156 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 $Q$ ...

**4**

votes

**3**answers

383 views

### Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...

**5**

votes

**0**answers

123 views

### Initial data and heat equation

We assume all solutions to be bounded here!
Let $y_{+},y_{-} \in C_c^{\infty}$ be two positive functions.
If we then consider the heat equation
$$\partial_t u(t,x) = \Delta u(t,x)$$ for two ...

**3**

votes

**1**answer

275 views

### Oscillatory integrals

Consider the integrals
$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$
I would like to know the asymptotic behavior of $I_n(\zeta,\epsilon)$ for ...

**1**

vote

**1**answer

50 views

### Limit of doubly indexed functions

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f_{n,j}$ converges pointwise a.e. ...

**1**

vote

**1**answer

55 views

### Can a sequence of degree one maps converges to a constant map in $W^{1,2}$ norm?

Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm?
If the answer is yes, would you please provide an explicit example?

**0**

votes

**0**answers

75 views

### Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...

**0**

votes

**1**answer

186 views

### Evaluation of a double definite integral with a singularity

How to compute the
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$
Is it possible to compute the integral analytically upto some terms. I believe it should involve ...

**1**

vote

**1**answer

82 views

### BV function with absolutely continuous divergence

Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...

**1**

vote

**1**answer

116 views

### Derivative and Jacobian determinant of solution of ODE [closed]

Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...

**2**

votes

**1**answer

103 views

### Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)

Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...

**0**

votes

**1**answer

48 views

### Boundary behavior of Greens functions on smooth bounded (planar) domains

It is well known that for any smooth bounded (connected) domain $\Omega\subset\mathbb R^d$ with $d\ge2$, we can define a Green's function $G:\Omega\times\Omega\to\mathbb R$ in $\Omega$ which is smooth ...

**2**

votes

**1**answer

73 views

### About the continuity of a function on BV

For a fixed $u \in BV(\mathbb{R}^N)$, consider the function
$h:(0,+\infty) \to BV(\mathbb{R}^N)$, given by
$h(t) = u (tx)$.
Is $h$ continuous?