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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Functions that are approximately differentiable a.e

The classical definition of an approximately differentiable function is as follows: Definition. Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
Piotr Hajlasz's user avatar
8 votes
2 answers
1k views

Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$. General question. Does there exist a simple expression for the ...
Max Alekseyev's user avatar
8 votes
1 answer
1k views

Traces of Sobolev spaces

Is there a simple proof of the following fact? Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\...
Piotr Hajlasz's user avatar
8 votes
3 answers
480 views

Invertibility of specific function

This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function: $$V\left(\varphi\right)=\frac{A\sqrt{\pi-...
Costas Vlachos's user avatar
8 votes
1 answer
359 views

A dichotomy for the quadratic variation of differentiable functions?

For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula $$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$ where the $\limsup$ is taken over all "partitions" ...
Iosif Pinelis's user avatar
7 votes
2 answers
419 views

On a monotonicity property of Fourier coefficients of truncated power functions

Is it true that $$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$ is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$? This question is related to this previous one. Twice integrating by parts, one ...
Iosif Pinelis's user avatar
7 votes
2 answers
1k views

Does anyone know what is the right reference for the following simple lemma from harmonic analysis?

The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds $$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
Changyu Guo's user avatar
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7 votes
1 answer
560 views

Hausdorff distance and Cauchy sequences

This is a generalization of an older question. Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > ...
Dominic van der Zypen's user avatar
7 votes
5 answers
2k views

Convexity of distance-to-boundary function

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ the distance-to-boundary function, that is, $$ d_{\Omega}\left(x\right):=\inf\left\...
Hadarmad's user avatar
7 votes
2 answers
264 views

Meeting a set of lines in $\mathbb{R}^n$

Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties? $M$ intersects all the elements of ${\...
Dominic van der Zypen's user avatar
6 votes
2 answers
588 views

Interpolation space between $L^1\cap L^2$ and $L^1$

In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
shrinklemma's user avatar
6 votes
1 answer
328 views

Inequality for functions on [0,1], continued

Let $0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set $$f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$$ Question. Is it true that, ...
Deepti's user avatar
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6 votes
1 answer
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Level sets of a Weierstrass nowhere-differentiable function

Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...
M Wright's user avatar
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6 votes
1 answer
172 views

Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?

Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated. Does there exist a sequence of ...
Asaf Shachar's user avatar
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6 votes
3 answers
2k views

A question on fractional derivatives

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...
user avatar
6 votes
3 answers
817 views

Mixtures of log-convex functions are log-convex: a reference

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
Iosif Pinelis's user avatar
5 votes
1 answer
924 views

Does a nonlinear additive function on R imply a Hamel basis of R?

A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
Keshav Srinivasan's user avatar
5 votes
2 answers
397 views

Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
Sascha's user avatar
  • 506
5 votes
1 answer
272 views

Is there any continuous ternary function which can not be represented by composition of continuous binary functions?

Let $f : X^3 \rightarrow X$. If $X$ is $\mathbb Z$, then there will be a couple of functions $g,h$ from $\mathbb Z^2$ to $\mathbb Z$ that satisfies $f(x,y,z) = g(h(x,y),z)$ since there is a bijection ...
damhiya's user avatar
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4 votes
1 answer
721 views

What is the dual space of $L^p$(conservative vector fields on a bounded set)?

First, some background: I wanted to prove that, if $f$ is a measurable function such that $\nabla f\in L^p_\text{loc}(\mathbb R^n)$, then $f\in L^p_\text{loc}(\mathbb R^n)$, $p\in(1,\infty)$. This is ...
Lentes's user avatar
  • 391
4 votes
3 answers
494 views

Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?

This is a follow-up on the previous question. Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb ...
Iosif Pinelis's user avatar
4 votes
2 answers
405 views

Smooth functions with zeros of infinite order on a closed set

It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(...
Longyearbyen's user avatar
4 votes
1 answer
1k views

Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $...
Hugo Chapdelaine's user avatar
4 votes
1 answer
200 views

Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
Sascha's user avatar
  • 506
4 votes
1 answer
942 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that $$\sum_{i=1}^...
Ktb's user avatar
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4 votes
0 answers
112 views

Find at least one square-boxed subcontinuum

Recall that a plane continuum is a closed, bounded, connected subset of the plane. It is non-degenerate if it contains at least two points. (We may sometimes just say "continuum" even if we ...
Mirko's user avatar
  • 1,345
4 votes
2 answers
233 views

On the monotonicity of the ratio of two logarithmic expressions

According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following: $r:=f/g$ is increasing on $(0,\pi/2)$ from $...
Iosif Pinelis's user avatar
4 votes
1 answer
394 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
James Propp's user avatar
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4 votes
1 answer
509 views

Derivatives of Riemann $\xi$ and traces of zeros

Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
Tom Copeland's user avatar
  • 9,937
4 votes
1 answer
323 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 ...
Giuseppe Negro's user avatar
3 votes
2 answers
2k views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
gondolier's user avatar
  • 1,829
3 votes
1 answer
279 views

$f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$

Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are strong measurable functions (they are the point wise a.e. limit of simple functions) and weak ...
Xiao's user avatar
  • 425
3 votes
4 answers
902 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
Mirko's user avatar
  • 1,345
3 votes
0 answers
646 views

On properties on a certain functional

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...
bambi's user avatar
  • 375
3 votes
1 answer
632 views

measure zero in R but not in R^2

I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
alich's user avatar
  • 33
3 votes
2 answers
182 views

Bounding integral expression with total variation of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$ for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
user avatar
3 votes
1 answer
424 views

Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
Axiom's user avatar
  • 520
2 votes
1 answer
923 views

Derivative and Jacobian determinant of solution of ODE [closed]

Let $\Phi$ be the unique solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ where we have assumed $f$ smooth. ...
user avatar
2 votes
0 answers
208 views

Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?

Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
Red shoes's user avatar
  • 359
2 votes
2 answers
239 views

Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$?

For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$, $$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}....
Wa haha's user avatar
  • 51
2 votes
0 answers
111 views

Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
2 votes
2 answers
326 views

Metrization of a topological vector space

Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ ...
user avatar
2 votes
0 answers
156 views

Are there hereditarily square-boxed plane continua?

A plane continuum is a bounded, closed and connected subset of the plane. A bounding box $B$ for a plane continuum $C$ is a rectangle $B=[a,b]\times[c,d]$ (including sides and interior) such that $C$ ...
Mirko's user avatar
  • 1,345
2 votes
0 answers
219 views

Integrating an n-fold Cauchy product of a Fourier series

I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here. Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
user363087's user avatar
2 votes
0 answers
936 views

On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality Let $A$ be a non-constant operator acting on $C^...
matilda's user avatar
  • 90
2 votes
0 answers
225 views

Dense property of intersection of Sobolev space

I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim: Pick an arbitrary real number $s$, we have that the ...
geooranalysis's user avatar
2 votes
2 answers
179 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
Iosif Pinelis's user avatar
2 votes
1 answer
373 views

Sturm Liouville problems for non-classical orthogonal polynomials

It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$ My ...
Amir Sagiv's user avatar
  • 3,544
2 votes
2 answers
460 views

Dual space of the completion of the space of Lipschitz functions

This question is a continuation of this post : Metrization of a topological vector space Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user avatar
1 vote
1 answer
250 views

Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ...
user550103's user avatar

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