The real-analysis tag has no wiki summary.

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### An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = ...

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42 views

### Relative homology of interlevel set

Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
...

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**2**answers

219 views

### Is there a reference for compact imbedding theory of Hölder space?

This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to ...

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**1**answer

241 views

### Kakeya and Nikodym maximal functions

I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...

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**1**answer

878 views

### Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...

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**1**answer

109 views

### Where find proof of such theorem about uniform convergence of differences

Where to find a proof of theorem which says that:
if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesque measure or on the set of second category with Baire propert ...

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**1**answer

168 views

### Ideals in a subalgebra in $C^\infty(M)$

Let $M$ be a (usual, finite dimensional) smooth manifold, and $C^\infty(M)$ the algebra of (real valued) smooth functions on $M$. For each point $a\in M$ and for each subalgebra $A$ in $C^\infty(M)$ ...

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265 views

### convergence of the infima of convex functions

Can one give a reference to a result like this:
If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then ...

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334 views

### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

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**4**answers

824 views

### A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...

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**1**answer

214 views

### Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...

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**1**answer

96 views

### To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function

I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...

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**1**answer

100 views

### Terminology for sequences/functions that approach each other

What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ...

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86 views

### Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?

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**1**answer

189 views

### When can we “displace” an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal ...

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**1**answer

99 views

### A recurrent sequence related to the Brouwer fixed-point theorem

Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} ...

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**1**answer

205 views

### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
...

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90 views

### A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis

I have a question. The question is to prove:
The weight $w \in A_\infty $if and only if
$\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all
...

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211 views

### A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on ...

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**0**answers

188 views

### The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...

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67 views

### Extending coverings over dense subsets

Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...

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123 views

### Does this sequence of H\"older functions have a limit?

Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
...

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167 views

### Proper Group action on a metric space

Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ ...

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236 views

### on completeness of R_mn, the set of all rational functions of type (m,n)

It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm.
Considering ...

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**7**answers

1k views

### Is Riemannian integration sufficient in physics?

Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration

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**2**answers

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### “Then obviously…” (a bound on strongly differential functions on an interval)

On the fourth page of their 1967 paper Local Behavior of Solutions of Quasilinear Parabolic Equations, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real ...

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**1**answer

123 views

### Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...

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438 views

### complete metric space

Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: ...

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288 views

### Polynomials and divided differences

I would greatly appreciate any hint for proving the following.
Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then ...

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**1**answer

114 views

### Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of ...

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42 views

### Question on weak Lorentz spaces

I have the following question. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. Suppose we have a function $f$ such that for $h>0$ it holds that
$$\|f\|_{H^1}\lesssim h^k.$$
By the Sobolev ...

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57 views

### Norm estimation of an area integral

I am solving a certain kind of integral equations using iteration and Volterra series. Now I get a formal solution and in order to prove convergence I need to estimate the $L^1$ and $L^2$ norm of the ...

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201 views

### Lipschitz map of the circle onto a triangle

Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane.

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241 views

### A question about “nice” functions

Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...

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335 views

### Is a Cauchy principal value invariant under a “change of variables”?

Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere ...

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**1**answer

164 views

### Lipschitz map of the ellipse

Is there a L-Lipschitz homeomorphism of the Elipse $x^2/4+y^2=1$ onto the unit circle $x^2+y^2=1$ such that $L<1$?

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410 views

### New differintegral formula: how is it related to other differintegral formulas?

Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to s} ...

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172 views

### The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...

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311 views

### Lower bounds on derivative around zero set of a positive smooth function

As part of a different problem, I came across the following simplified question, for which I cannot exhibit a proof nor a counterexample. Note that the assumptions of smoothness and strict positivity ...

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58 views

### a question on bounds for complex functions

Let $\bf{u}$ be a smooth complex vectorfield defined on the closed unit ball $B_1(0)\subset \mathbb{R}^3$. Let $\phi$ be any smooth complex function defined on $B_1(0)$.
My question is, is the ...

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435 views

### Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I_n
:=
...

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### Literature on Exponential of a Quadratic Form

Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
...

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377 views

### Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int_a^b ...

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### Adjoint operator in sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta ...

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### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...

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### Does the Border (Boundary) Points of a convex body make a concave function?

Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis ...

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### Last Point of Exit from the 2-D Positive Quadrant

I define the functions $f_i(\mathbf{u}),i=1,2$ and $g_i(\mathbf{u}),i=1,2$ where $\mathbf{u}\in\mathbb{C}^{N}$ is the unit-norm vector. Thus, this functions are defined over the unit norm $N-$sphere. ...

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218 views

### When does the finite union of convex sets have a hole in it?

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) ...

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172 views

### Boundedness of an Oscillating Integral

Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded as ...

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288 views

### Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series.
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...