Questions tagged [real-analysis]

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

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Necessary and Sufficient conditions for integrable function [closed]

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^\infty ...
Xiaopai Song's user avatar
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Proving that $\|\mathbf{T}^n\|^2=\sum_{g\in \mathbf{G}(n,d)}\|\mathbf{T}_g\|^2\,$

Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$. For ${\bf A} = (A_1,...,A_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by $...
Schüler's user avatar
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1 answer
111 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 \...
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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 $g$...
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2 answers
184 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 ...
co.sine's user avatar
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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 ...
Ben Curnow's user avatar
5 votes
3 answers
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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 ...
Y.B.'s user avatar
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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) $$ U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t $$ ...
SerkanSüner's user avatar
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Uniformly Bounded (updating)

Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ 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{...
Xiaopai Song's user avatar
11 votes
2 answers
745 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 ...
James Baxter's user avatar
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3 votes
2 answers
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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 ...
Y.B.'s user avatar
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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 \...
Y.B.'s user avatar
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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 $$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$ ...
random_person's user avatar
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2 answers
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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!
James Baxter's user avatar
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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 ...
James Baxter's user avatar
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1 answer
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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 $\|\cdot\|$ ...
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3 votes
1 answer
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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 ...
Oliver Kullmann's user avatar
2 votes
1 answer
457 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 \...
Richard's user avatar
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7 votes
1 answer
280 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 ...
D. Driggs's user avatar
5 votes
0 answers
254 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 ...
Elliott's user avatar
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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 ...
Max's user avatar
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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-...
Jean's user avatar
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1 answer
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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)...
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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 $\|\...
Idonknow's user avatar
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7 votes
1 answer
454 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 ...
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5 answers
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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 ...
John Bentin's user avatar
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5 votes
1 answer
759 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. ...
James Baxter's user avatar
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0 votes
4 answers
546 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 ...
luw's user avatar
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5 votes
1 answer
190 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'\|...
Jairo Bochi's user avatar
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2 votes
1 answer
463 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)-...
user avatar
2 votes
1 answer
163 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} ...
T.Sell's user avatar
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3 votes
0 answers
382 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|\; \...
James Baxter's user avatar
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5 votes
1 answer
578 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 ...
James Baxter's user avatar
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0 votes
1 answer
136 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....
James Baxter's user avatar
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0 votes
1 answer
194 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 ...
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0 votes
1 answer
116 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?
James Baxter's user avatar
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-5 votes
1 answer
182 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 ...
Haoyu Wang's user avatar
16 votes
1 answer
2k 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)) \...
James Baxter's user avatar
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9 votes
0 answers
293 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 ...
Mason's user avatar
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1 vote
0 answers
166 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{\...
VVCM's user avatar
  • 111
2 votes
0 answers
276 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(...
grge's user avatar
  • 21
3 votes
1 answer
285 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 ...
Markus Holzleitner's user avatar
1 vote
0 answers
76 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 ...
Pete Caradonna's user avatar
8 votes
1 answer
263 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 $$Of(x) = \lim _{d \to 0} \sup _{y, z \in B_d (x)} |f(y) - f(z)| \, .$$ ...
James Baxter's user avatar
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1 vote
0 answers
110 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 $k$-...
Ramen's user avatar
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0 votes
1 answer
128 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-...
Testform's user avatar
5 votes
1 answer
573 views

Extending continuous functioms defined on the irrationals

Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G_\delta $ subset of $\...
Ruy's user avatar
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0 votes
0 answers
91 views

Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$. If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user avatar
2 votes
1 answer
133 views

The blow-up rate of a nonlinear oscillator

(Related to this Math.SE question.) For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$ ...
Giuseppe Negro's user avatar
1 vote
1 answer
118 views

Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?

This is the question that I should have asked before asking this older question. If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ ...
Dominic van der Zypen's user avatar

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