# Questions tagged [real-analysis]

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

2,472 questions

**2**

votes

**1**answer

33 views

### Necessary and Sufficient conditions for integrable function

Suppose that $a, b$ and $c$ are constant. Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int_0^{\infty}\int_0^{\infty}\int_0^{\...

**1**

vote

**1**answer

64 views

### Continuous inclusion of metric spaces of smaller capacity

If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...

**0**

votes

**0**answers

97 views

+100

### How do we introduce a signed finite measure on the space of curves confined into the box $[0,1]^{n}$?

Given $\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$, consider the equivalence relation:
\begin{align*}
& \alpha_{1} \sim \alpha_{2} \Leftrightarrow \int_{0}^{1}...

**0**

votes

**0**answers

45 views

### Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation
\begin{align}
g(x)= \int f(t) h(t,x) dt
\end{align}
such that $f$ and $g$...

**1**

vote

**2**answers

118 views

### One-Sided Analyticity Condition Guarantees Analytic Function?

Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:
$f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$.
$f$ satifsfies the ...

**3**

votes

**0**answers

120 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 ...

**4**

votes

**3**answers

136 views

### Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...

**0**

votes

**1**answer

186 views

### Stone–von Neumann theorem?

The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...

**1**

vote

**1**answer

250 views

### Uniformly Bounded (updating)

Suppose that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2,$ and $b_1, b_2>0$. For $x,y>0,$
(1) define a fucntion
$$H(x,y)=\frac{x^{\frac{1}{2}}\...

**7**

votes

**2**answers

209 views

### Textbook recommendation request: Exercises to supplement Evans and Gariepy

While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it ...

**2**

votes

**2**answers

116 views

### Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...

**2**

votes

**1**answer

65 views

### Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...

**-1**

votes

**0**answers

76 views

### Proof that $2^y x + 2^y - 1$ is an closed formula of $g(1,x,y)$ [closed]

I should proof that $2^y x + 2^y - 1$ is an closed formula of $g(1,x,y)$ with induction or something else.
Given is:
$g(n,x,y)=\begin{cases} x+y,\quad if\quad n=0 \\ x,\quad \quad \quad if\quad n>...

**8**

votes

**2**answers

822 views

### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

**1**

vote

**2**answers

159 views

### Reference request: Functions of bounded variation in one real variable

Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!

**0**

votes

**0**answers

18 views

### Parametric statistics: how to estimate the supremum of a set of parameters from a random sample

I would like to ask a question on how to estimate the supremum norm of a set of parameters in the following setting. I appreciate any pointer or suggestion. Thanks.
Question:
Suppose we have $m$ ...

**-1**

votes

**0**answers

43 views

### Changing width of integration [on hold]

For $F: \mathbb{R} \to \mathbb{R}$ and $w \in \mathbb{R}$, define $g_w(a) = f(a + w) - f(a)$.
Given $g_1$ (and not $F$), how can we compute $g_w$ for other $w$?
We can assume $F$ is well behaved: $...

**1**

vote

**2**answers

80 views

### Quantitative bound on irrational rotation recurrence time

Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb ...

**0**

votes

**1**answer

88 views

### Smallest Lipschitz Constant of a Differentiable Function [closed]

Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ ...

**3**

votes

**1**answer

188 views

### A certain generalisation of the golden ratio

Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...

**1**

vote

**1**answer

149 views

### Approximation of a two-variable function by tensor products

Let $X$ and $Y$ be compact metric spaces and $f: X \times Y \to \mathbb{R}$ be a continuous function.
We know that, for every $n \in \mathbb{N}$, by the Stone-Weierstrass theorem, there exist $k_n \...

**7**

votes

**1**answer

157 views

### Almost orthonormal projection and orthonormal projection in Hilbert space

Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...

**5**

votes

**0**answers

135 views

### Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...

**0**

votes

**1**answer

60 views

### Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...

**4**

votes

**0**answers

79 views

### On a much weaker version of the Normal conjecture

I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...

**3**

votes

**1**answer

96 views

### What “mild solution” means, and how to find it?

In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem (1)...

**0**

votes

**0**answers

73 views

### If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...

**6**

votes

**1**answer

131 views

### Anisotropic perimeter and regularity of anisotropic minimal surfaces

1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...

**-1**

votes

**0**answers

58 views

### Is this expression true? [migrated]

Let $a_1=b_1/h,...,a_n=b_n/h\in\mathbb{R}$ with $h\in\mathbb{R}$ small. It's true that, given a $\alpha\in\mathbb{R}$:
\begin{eqnarray}
(a_1+...+a_n)^\alpha=\sum_{i=1}^n (a_i)^\alpha+\mathcal{O}\left(\...

**22**

votes

**4**answers

713 views

### Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products?

The question below was posted on Mathematics Stack Exchange. It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I wonder ...

**5**

votes

**1**answer

247 views

### Functions that map open balls to open balls of different radius?

For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.
...

**0**

votes

**4**answers

319 views

### How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$ [closed]

How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this ...

**0**

votes

**0**answers

54 views

### Prove that an iterative estimate implies Holder continuity

Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...

**5**

votes

**1**answer

130 views

### An inequality involving $L^1$ and $L^\infty$ norms of a function of a real variable and its derivative

I got to the following inequality by a (hopefully correct) tortuous argument:
If $F:[a,b] \to \mathbb{R}$ is a absolutely continuous monotone function then:
$$ \|F'\|_1^2 \leq 4 \|F\|_1 \, \|F'\|...

**2**

votes

**1**answer

107 views

### Difference quotient for functions of bounded variation

Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...

**2**

votes

**1**answer

85 views

### Approximate sequence of numbers

Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers
$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$
It is easy to see that these numbers satisfy
$$x_{n,0} = \frac{1}{n+1} ...

**3**

votes

**0**answers

347 views

### What tools from functional analysis are relevant to investigating this operator?

Given a sequence of continuous functions ${{f_n}}$, define the varicontinuity index $$V({f_n}): \mathbb{R} \to [0, \infty]$$ by
\begin{split}
V({f_n})(x) &=\sup \Big\{\varepsilon > 0\big|\; \...

**5**

votes

**1**answer

521 views

### A problem in real analysis of a topological nature

Let $f: R \to R$ be a function such that the closure of its graph contains as a subset the graph of a uniformly continuous function. Does there exist a dense subset $S$ of $R$ such that the ...

**0**

votes

**1**answer

110 views

### Given these conditions, can a function be defined that is well defined a.e.?

I have two functions, and I want to combine them to define a certain function.
Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a....

**0**

votes

**0**answers

41 views

### Limiting a sequence of moment generating functions [migrated]

I was trying to solve the following problem:
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...

**0**

votes

**1**answer

64 views

### Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$

Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...

**0**

votes

**1**answer

88 views

### Are these conditions enough to ensure joint measurability?

Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?

**-5**

votes

**1**answer

115 views

### a question of definite integral [closed]

1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$
2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$
please help me. I tried to use MATLAB but ...

**14**

votes

**1**answer

1k views

### Are continuous functions almost completely determined by their modulus of continuity?

Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by
$$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \...

**2**

votes

**0**answers

133 views

### a Kernel free asymptotic for a sampling operator

Let $\Pi=\left\{ t_{k}\right\} _{k\in\mathbb{Z}}$ a sequence of real numbers such that $-\infty<t_{k}<t_{k+1}<+\infty$ for every $k\in\mathbb{Z}$, $\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty$...

**5**

votes

**0**answers

100 views

### Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...

**1**

vote

**0**answers

56 views

### On the convergence of an integral of Hardy's maximal function

Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function.
Assume that
$$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ ...

**1**

vote

**0**answers

41 views

### Singular integral of the composition of the Hilbert transform and fractional Laplacian

Given $0<s<1$, we can define the Fractional Laplacian by
$$\Lambda^{-s}f(x):=(-\Delta)^{-s/2}(x)=\int_{-\infty}^{+\infty}|x-y|^{-1+s}f(y)dy$$
or by means of Fourier transform as $$\widehat{\...

**2**

votes

**0**answers

110 views

### Prove conjugation law for Exp(z) using only Exp(x + y) = Exp(x)Exp(y) [closed]

Starting with just the property $E(x + y) = E(x)E(y)$, one can prove quite a lot of the main properties of the exponential function on real numbers. For example, $E(0) = 1$, and $E'(X) = E(X)$, and $E(...

**3**

votes

**1**answer

112 views

### Kantorovich duality with pseudometrics

The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that ...