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
1answer
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.
$$\int_0^{\infty}\int_0^{\infty}\int_0^{\...
1
vote
1answer
64 views
Continuous inclusion of metric spaces of smaller capacity
If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...
0
votes
0answers
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
0answers
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
2answers
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
0answers
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
3answers
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
1answer
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
1answer
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
$$H(x,y)=\frac{x^{\frac{1}{2}}\...
7
votes
2answers
209 views
Textbook recommendation request: Exercises to supplement Evans and Gariepy
While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it ...
2
votes
2answers
116 views
Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
2
votes
1answer
65 views
Uniqueness of minimizers in the Calculus of Variations
Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...
-1
votes
0answers
76 views
Proof that $2^y x + 2^y - 1$ is an closed formula of $g(1,x,y)$ [closed]
I should proof that $2^y x + 2^y - 1$ is an closed formula of $g(1,x,y)$ with induction or something else.
Given is:
$g(n,x,y)=\begin{cases} x+y,\quad if\quad n=0 \\ x,\quad \quad \quad if\quad n>...
8
votes
2answers
822 views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
1
vote
2answers
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
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
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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} $$
and $\alpha$ is ...
5
votes
0answers
135 views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
0
votes
1answer
60 views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
4
votes
0answers
79 views
On a much weaker version of the Normal conjecture
I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
3
votes
1answer
96 views
What “mild solution” means, and how to find it?
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem (1)...
0
votes
0answers
73 views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
6
votes
1answer
131 views
Anisotropic perimeter and regularity of anisotropic minimal surfaces
1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...
-1
votes
0answers
58 views
Is this expression true? [migrated]
Let $a_1=b_1/h,...,a_n=b_n/h\in\mathbb{R}$ with $h\in\mathbb{R}$ small. It's true that, given a $\alpha\in\mathbb{R}$:
\begin{eqnarray}
(a_1+...+a_n)^\alpha=\sum_{i=1}^n (a_i)^\alpha+\mathcal{O}\left(\...
22
votes
4answers
713 views
Is this a known question about the expression of a function on $\Bbb R^2$ as an infinite sum of products?
The question below was posted on Mathematics Stack Exchange. It received no answer, and I do not expect any direct answer to it here. However, the question seems to me a natural one. Thus I wonder ...
5
votes
1answer
247 views
Functions that map open balls to open balls of different radius?
For $n \geq 2$ we say a continuous function $f: \mathbb R^n \to \mathbb R^n$ such that the image of any bounded open ball is a bounded open ball of different radius is a balloon function.
...
0
votes
4answers
319 views
How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$ [closed]
How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $C_k$ is the catalan number: $C_k=\frac{1}{k+1}{2k \choose k}$. (Further, is there any general method to treat this ...
0
votes
0answers
54 views
Prove that an iterative estimate implies Holder continuity
Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...
5
votes
1answer
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:
$$ \|F'\|_1^2 \leq 4 \|F\|_1 \, \|F'\|...
2
votes
1answer
107 views
Difference quotient for functions of bounded variation
Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
2
votes
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]$$ by
\begin{split}
V({f_n})(x) &=\sup \Big\{\varepsilon > 0\big|\; \...
5
votes
1answer
521 views
A problem in real analysis of a topological nature
Let $f: R \to R$ be a function such that the closure of its graph contains as a subset the graph of a uniformly continuous function. Does there exist a dense subset $S$ of $R$ such that the ...
0
votes
1answer
110 views
Given these conditions, can a function be defined that is well defined a.e.?
I have two functions, and I want to combine them to define a certain function.
Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a....
0
votes
0answers
41 views
Limiting a sequence of moment generating functions [migrated]
I was trying to solve the following problem:
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables with the probability mass function $P\{X_n = \pm1 \} = \frac{1}{2}$, $n \in \...
0
votes
1answer
64 views
Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$
Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...
0
votes
1answer
88 views
Are these conditions enough to ensure joint measurability?
Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?
-5
votes
1answer
115 views
a question of definite integral [closed]
1.$$\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
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
0answers
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
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
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
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 ...
