Skip to main content

All Questions

Filter by
Sorted by
Tagged with
0 votes
1 answer
224 views

Special functions on the unit disk

Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk. We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following: 1) it is a ...
expmat's user avatar
  • 1,271
8 votes
1 answer
2k views

Does integrating with respect to a finitely additive measure respect addition?

Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we ...
Daniel Barter's user avatar
1 vote
1 answer
6k views

How to determine whether a multivariate function is bounded or not

Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not? For example, there is a function $f(x,y)=-x^2+...
Allen's user avatar
  • 141
2 votes
2 answers
711 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
Richard Hevener's user avatar
2 votes
1 answer
2k views

Modified Lebesgue differentiation theorem

Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
Florian's user avatar
  • 2,270
9 votes
1 answer
10k views

Can the supremum of continuous functions be discontinuous on a set of positive measure? [closed]

Given a sequence of continuous functions $f_n(x)$, all defined on a compact set $D$ and assuming $f_n(x)$ is uniformly bounded. Let $f(x) = sup_n f_n(x)$. It is clear that $f(x)$ is not necessarily ...
user18629's user avatar
3 votes
0 answers
211 views

Elementary analysis: reference request

Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$: $T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$ So essentially ...
Tom Ellis's user avatar
  • 2,895
2 votes
0 answers
224 views

Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental [closed]

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts. At page $6$, the ...
C.S.'s user avatar
  • 4,795
4 votes
2 answers
1k views

$L^1$ norm of the Fourier transform of a truncated Gaussian

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes: Consider the Gaussian $G(x):=e^{-x^2}$ ...
user17240's user avatar
  • 852
21 votes
0 answers
1k views

Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$

A while ago, I came across the following problem, which I was not able to resolve one way or the other. Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
George Lowther's user avatar
5 votes
2 answers
560 views

implicit function theorem for algebraic sets

We know by the standard Implicit Function Theorem that If $f:\mathbb R^4\rightarrow\mathbb > R^2$ is a polynomial (or in fact any continuously differentiable function), then there is a ...
filipm's user avatar
  • 1,359
1 vote
1 answer
224 views

Can symmetrizing a contraction increase the speed of convergence?

Dear community, I have a problem which is very simple to state but seems to be hard to answer. Statement of the problem Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
herrsimon's user avatar
  • 199
6 votes
2 answers
929 views

reverse mathematics strength of "Lipschitz functions are somewhere differentiable"

What is the reverse mathematics strength of "For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ? (...
user avatar
0 votes
0 answers
165 views

minimizing the integral of a function over square sets.

Hi! I'm interested in some problems, but to be honest i'm not sure of the field they belong to. Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
kaleidoscop's user avatar
  • 1,352
2 votes
1 answer
942 views

A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\...
user7738's user avatar
  • 173
67 votes
9 answers
7k views

Taking "Zooming in on a point of a graph" seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
Steven Gubkin's user avatar
3 votes
3 answers
522 views

Closure of singular points

Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular form. $$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y + \frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + \frac{f_{40}}{...
Ritwik's user avatar
  • 3,245
9 votes
2 answers
2k views

Does the Weierstrass function have a point of increase?

Problem The Weierstrass function $W(x)$ is given by $W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$ where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$. A function $f:\mathbb{R}\...
Bati's user avatar
  • 491
1 vote
1 answer
420 views

density of a set

let $S=\{\sin (n)|n \in N\}$. We can prove $S$ is dense in $[-1,1]$. So is the set $\{\sin( n^2)|n \in N\}$; but the set $\{\sin (n^3)| n \in N\}$ is not dense in $[-1,1]$. How to prove this?
gubo's user avatar
  • 11
-3 votes
1 answer
332 views

Convergence Question [closed]

If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
David's user avatar
  • 1
6 votes
2 answers
2k views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
q.g's user avatar
  • 71
11 votes
6 answers
18k views

One-line proof of the Euler's reflection formula

A popular method of proving the formula is to use the infinite product representation of the gamma function. See ProofWiki for example. However, I'm interested in down-to-earth proof; e.g. using the ...
juno's user avatar
  • 111
4 votes
2 answers
323 views

Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?

Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that $$\lim_{n} ~a_{nn} = L_1, \qquad \lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$ exist. Is there a sufficient criteria to conclude ...
Ritwik's user avatar
  • 3,245
10 votes
1 answer
1k views

Real analytic function, injective, non surjective and preserving the rationals ?

I'd like to prove the non-existence of a real analytic function, injective, non-surjective that sends rationals to rationals. Is it a classical result ? If not, any hints on how to prove it ? Thanks ...
christian aebi's user avatar
2 votes
3 answers
913 views

A definite integral

Hello, I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...
Anand's user avatar
  • 1,649
6 votes
2 answers
2k views

Continuity of a convolution (Version 2)

Hello, This problem bothers me for some time. Suppose that $\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support); $\psi$ is ...
2 votes
2 answers
2k views

Does a bounded real function have an analytic continuation [closed]

Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where $f$ is real-analytic on the open interval $(0,1)$ $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-...
Essex's user avatar
  • 23
2 votes
1 answer
276 views

Conformal Extension from a closed set to open

Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
zapkm's user avatar
  • 541
1 vote
2 answers
1k views

An interesting doubly infinite series

Let $0<\mu<1$ and $\alpha:=1-\mu^2$. Consider the function $$f(x):=x\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}x}-\frac{1}{x}\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}/x},$$ ...
Erwin's user avatar
  • 201
3 votes
0 answers
237 views

Monotonicity of a certain parametric integral

I would like to ask for some help (hints, ideas) in solving the following problem: Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that if we define for any $x\in\mathbb{R}...
Maciej Skorski's user avatar
1 vote
1 answer
528 views

Function space between uniform continuity and Hölder continuity

Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder ...
shuhalo's user avatar
  • 5,327
1 vote
0 answers
346 views

Gauge integral of the derivative of a function except on a set of measure 0.

For the entire question, the interval I am integrating over is $[0,1]$. Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some ...
Hunter Spink's user avatar
5 votes
1 answer
543 views

Acceleration via smoothing

Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature? Let $f(t)=(\sum_{1 \leq n \leq ...
James Propp's user avatar
  • 19.7k
1 vote
2 answers
382 views

A question about zeros of Tate type integral

Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $...
user1832's user avatar
  • 2,709
8 votes
2 answers
471 views

Multiplying functions on the unit square as generalized matrices

Consider the $\mathbb{R}$-vector space of sufficiently nice real-valued functions on the unit square $I^2$, where "sufficiently nice" could be taken to mean any one of a number of things - say ...
Bruno Joyal's user avatar
  • 3,910
4 votes
1 answer
354 views

Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
Harry Altman's user avatar
  • 2,585
4 votes
1 answer
561 views

Taylor Series Remainder

Suppose I have a $C^\infty$ smooth function $f$ defined on the reals. I can apply Taylor's formula and get the local expression $$ f(x) = \sum_{i=0}^l\frac{f^{(i)}(0)}{l!}x^i+ f^{(l+1)}(\xi(x))x^{l+...
Philipp's user avatar
  • 979
1 vote
1 answer
3k views

Is point to set distance continuous?

Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
Maj's user avatar
  • 27
4 votes
1 answer
1k views

Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
Feldmann Denis's user avatar
4 votes
1 answer
306 views

ordered fields with the bounded value property, without choice

In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ ...
James Propp's user avatar
  • 19.7k
15 votes
1 answer
2k views

Interpolating between piecewise linear functions, with a family of smooth functions

Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that: there is a number $k\in\mathbb N-\{0\}$ ...
Cristi Stoica's user avatar
5 votes
1 answer
878 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
Tom LaGatta's user avatar
  • 8,512
61 votes
1 answer
5k views

Every real function has a dense set on which its restriction is continuous

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous. Or so I'm told, but this leaves me ...
Gro-Tsen's user avatar
  • 32.5k
4 votes
2 answers
2k views

a different nested intervals theorem

Is there any literature on (and a standard name for) the proposition that for any arbitrary-cardinality collection of closed intervals in the reals that is nested (in the sense that, given any two of ...
James Propp's user avatar
  • 19.7k
4 votes
1 answer
222 views

a closed-form for mean/integral, but weighting positive differences between values and "mean" differently from negative differences?

Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\...
matt j's user avatar
  • 41
2 votes
2 answers
643 views

Estimating the Hausdorff measure of a subset of the sphere

Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these ...
Florian's user avatar
  • 2,270
3 votes
1 answer
1k views

ordered fields with the bounded value property

Say that an ordered field $F$ satisfies the bounded value property if, for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there ...
James Propp's user avatar
  • 19.7k
8 votes
3 answers
785 views

truth vs. provability for ordered fields

In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
James Propp's user avatar
  • 19.7k
8 votes
4 answers
3k views

Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
Harry Altman's user avatar
  • 2,585
6 votes
2 answers
812 views

A dual theory to the theory of currents?

The k-currents are defined as dual space to the spaces of all smooth k-forms. (These monsters are used to work with the minimal k-surfaces.) Assume I want to look at the generalized k-forms; they can ...
ε-δ's user avatar
  • 1,785

1
108 109
110
111 112
114