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Questions tagged [real-analysis]

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

2
votes
0answers
91 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 ...
2
votes
2answers
63 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
1answer
163 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) $$ U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t $$ ...
0
votes
1answer
150 views

Uniformly Bounded

If $a_1<1$, $a_1+a_2+a_3>1$, for $x,y,z>0,$ (1) define a fucntion $$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1} (1+t)^{a_2+1} (1+t+z)^{a_3}}\exp\big\{-\frac{x}{1+t}-\...
7
votes
2answers
200 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
2answers
109 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 ...
1
vote
1answer
54 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
0answers
74 views

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

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>...
7
votes
2answers
803 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
2answers
150 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
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0answers
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
0answers
42 views

Changing width of integration

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
2answers
78 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
1answer
85 views

Smallest Lipschitz Constant of a Differentiable Function [on hold]

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
1answer
187 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
1answer
141 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
1answer
152 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 ...
-2
votes
0answers
58 views

Spectrum and operators [closed]

Hwo can help me to prove that if the spectrum of a normal operator lies on a circle {z∈C:∣z∣=1}, then this operator is unitary. I'm very thankful for any ideas and help
4
votes
0answers
109 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
1answer
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
0answers
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
1answer
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
0answers
69 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
1answer
130 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
0answers
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
4answers
702 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
1answer
246 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
4answers
317 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
0answers
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
1answer
129 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
1answer
105 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
1answer
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
0answers
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
1answer
518 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
1answer
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
0answers
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
1answer
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
1answer
87 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
1answer
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
1answer
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
0answers
132 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
0answers
95 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
0answers
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
0answers
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
0answers
109 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
1answer
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 ...
1
vote
0answers
36 views

Potential for a Monotone Operator

[Cross-posted from math.stackexchange] I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
7
votes
1answer
188 views

Oscillation operator of a function

Call a function from $[0, 1]$ to itself a box function. Given any box function $f$, define its oscillation function $Of$ as $$Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)| \, .$$ ...
1
vote
0answers
83 views

Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$

I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently: Let $X$ be a compact $k$-...
0
votes
1answer
93 views

Can we find the $a$ value?

We have the following limit with and $a \in \mathbb{R}$ and $ u \in \mathbb{R}$ . And here, ${\lfloor x \rfloor}$ is floor function $$\lim_{u \rightarrow \infty} \frac{f(a)-\int_1^u ( {x-...