Tagged Questions
0
votes
1answer
32 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
4
votes
4answers
89 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
0
votes
0answers
17 views
Multiplication and Division of functions
Suppose that you have two continuous functions, $f(x)$ and $g(x)$.
Suppose that you have numerical approximations for these functions, stored a vectors, $f^*$ and $g^*$.
If I want to approximate ...
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
0
votes
1answer
37 views
A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
0
votes
2answers
44 views
A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
0
votes
3answers
38 views
Can there be a non-polynomial continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has multiple zero-valued points?
1) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ that has more than one zero-valued point in domain?
2) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ ...
0
votes
0answers
68 views
Can a function with finite discontinuities be nicely approximated by a continuous function?
1) Can every discrete function be approximated by some continuous function with regards to domain defined?? (By discrete function, I mean: there is countably infinite number in domain, and for each ...
1
vote
0answers
22 views
A modular which is not a metrizing modular (hence not an F-norm)?
I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces.
Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
6
votes
1answer
138 views
Exercise Functional Analysis
Let $\mathcal{F}$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$.
Consider an operator $\mathcal{O}: \mathcal{F} \rightarrow \mathcal{F}$ such that:
$\mathcal{O}( f_1 + f_2) = ...
2
votes
1answer
35 views
Cross Product for functions
So functions are just uncountabley-infinite dimensional vectors, and as such there's a nice generalization of the inner product between two functions (the integral of their product). Is their a ...
4
votes
1answer
80 views
Is this matrix function convex or non-convex?
Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
3
votes
3answers
75 views
Periodic polynomial?
I was thinking if it was possible to create a polynomial that would be periodic all over the reals, since polynomials can be periodic on an interval. I then I found out the following function:
...
0
votes
1answer
55 views
How to come up with a function?
Say, I have a hypothetical value. And it increases at start and after some point it decreases (decays) at constant rate (average) and it tends to 0. It looks something like a Poisson distribution.
...
1
vote
2answers
50 views
Showing that a modifying function which is continuous at 0 is uniformly continuous
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
0
votes
0answers
61 views
functional equation, how to solve
Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...
1
vote
1answer
45 views
Showing that a function is a modifying function (how to prove subadditivity)
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
2
votes
2answers
63 views
A question concerning measurability of a function
Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on ...
3
votes
1answer
73 views
Application of Banach fixed-point theorem
I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$.
The first thing I do is to define a function ...
1
vote
0answers
53 views
Continuity and openness of the map $C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$
I need to prove or disprove that the composition operator is continuous and open. Consider the following map
$$h:C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$$
that takes a function ...
6
votes
2answers
114 views
equivalence of norms
I would like a little help here:
I have two defined norms over $C^{1}([0,1])$ :
$\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$
$\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$
I already ...
2
votes
1answer
52 views
Positive Continuous functions tending to $0$
Let $f(x) > 0$ be a member of $C(a, {\infty})$, that is, the space of continuous functions from the real number a to $+{\infty}$ . Suppose further that $f$ tends to $0$ as $x$ tends to $+{\infty}$.
...
0
votes
1answer
44 views
Norm of normal Operator A
I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$
My question: What exactly means $\sigma(A)$ and why this is true ?
I always thouht the only way to get the ...
2
votes
2answers
69 views
Inverse of inverse of function?
What is the inverse of inverse of a function (I assume the original function is invertible)? Is this the original function? Is it always true?
1
vote
1answer
36 views
Function for unique hash code
I am interested in finding $F(x,y)$, such that $x$ and $y$ $\in \mathbb Z^+$ and $F(x,y)$ is one to one function i.e., $F(x,y)$ is unique for any unique unordered pairs of $x$ and $y$.
Regards,
...
0
votes
0answers
38 views
inclusion of $C^{\infty}$ to B-spaces $L^p$
if the function is $C^{\infty}(D)$, does it imply that this function is also in all of the $L^p$ spaces as well?
Edit: I am looking at $u_t=x^2u_{xx}+yu_y, \; D=[0,1]\times[0,1]\times [0,T]$. I can ...
3
votes
2answers
123 views
Infinite sum of floor functions
I need to compute this (convergent) sum
$$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$
But I have no idea how to get rid of the floor thing. I thought ...
3
votes
1answer
86 views
question about the definition of linear functions/operators (domains)
Suppose $\Omega_s \subset \mathbb{R}^n$ is a compact subset for each $s \in [0,T]$. I have a linear operator
$$p_t^s:H^1(\Omega_t) \to H^1(\Omega_s)$$
which maps functions on $\Omega_t$ to functions ...
3
votes
1answer
168 views
Minimising the maximum distance between bounded functions for a given bounded function
Let $g\in C[0,1]$ where $C[0,1]$ is the vector space of continuous functions on [0,1].
For any function $f$, define $A_f$ $= \sup\{\sqrt{(x-y)^2+(f(x)-g(y))^2}:x,y\in[0,1]\}$
Can the "sup" be ...
1
vote
1answer
113 views
Functional independence
Definition confusion:
I wish to show that $$f(x,y)={-y\over x}$$ and $$g(x,y)=\log |x|$$ are functionally independent on some domain.
What does that mean? What do I have to show? And how does one ...
2
votes
1answer
58 views
If f is in LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b].
Question:
If f $\in$ LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b].
Context: f $\in$ LipK[a,b] then it is Lipschitz with constant K.
The text I am currently ...
1
vote
1answer
54 views
orthonormalsystem but not orthonormalbasis
I would like to know how to show that the function
$r_n(t)=\mbox{sgn}\big(\sin(2^n \pi t)\big)$, sgn = sign function
is an orthonormal system but not an orthonormal basis from $L_2([0,1])$
3
votes
2answers
235 views
Extreme points of unit ball in $C(X)$
Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuos functions in sup-norm.
I read in Douglas' Banach algebra techniques in operator theory that the followings are equivalent:
...
1
vote
0answers
23 views
Series function help
I want to find a function such that $$ \sum_{0<j<n/k
} f(kj)=1 $$
Where the sum j is taken over the natural numbers,
And the series is satisfied for all integers k and n, I was thinking of ...
0
votes
1answer
102 views
Find all positive functions of a positive real such that $ f(xf(y))=yf(x) $ and $\lim_{x\to\infty}f(x)=0$
Find all functions defined on the set of positive reals which take positive real values and satisfy:
$$ f(xf(y))=yf(x) $$
for all ; $ f(x)\to0 $ and as $ x\to\infty $
37
votes
2answers
537 views
Looking for a function such that…
There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is:
...
1
vote
0answers
51 views
if the system localized
Let g be integrable, even function. Let $\Lambda$ be uniformly discrete sequaence (i.e. $\inf|\lambda_i-\lambda_j|>0, \lambda_i, \lambda_j \in \Lambda$).
We say that system $\Psi=\{\psi_n\}$ is ...
12
votes
2answers
520 views
isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?
Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$.
Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
2
votes
2answers
107 views
Positive twice differential decreasing function, is it convex?
If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
1
vote
2answers
262 views
Sequence of Uniformly Bounded functions
Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$.
Assume the following.
For any sequence $\{X_k\}_{k=1}^{\infty}$ of ...
1
vote
0answers
64 views
Is there chance to form a frame (Riesz basis)?
Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions
$$
f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)
$$
One can show that ...
14
votes
2answers
333 views
Increasing orthogonal functions
What is the maximal $n$ such that there exist functions $f_1, \dots, f_n:[0,1] \to \mathbb{R}$ that are all bounded, non-decreasing, and mutually orthogonal in $L^2([0,1])$?
2
votes
0answers
208 views
$L_2$-norm representation of the function
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$(see for reference ...
3
votes
5answers
1k views
Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?
It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.
2
votes
1answer
148 views
Periodic functions composition
could you help me with following problem?
I need to find two non-periodic functions "f" and "g" where their composition f(g) will be periodic.
Note that constant function is periodic too.
Thanks ...
1
vote
1answer
131 views
find a mapping of two probability mass function to another probability mass function
Let $p$ and $q$ be arbitrary probability mass functions of two discrete random variables. I need examples of functions $F(p,q)$ such that $r = F(p,q)$ and $r$ is a probability mass function for some ...
3
votes
0answers
147 views
Topology of bijective functions between Banach spaces
Suppose $X,Y$ are Banach spaces and consider the space $C(X,Y)$ of bounded continuous functions $X \to Y$ with the supremum norm. Are there any results about the the topological properties of the set ...
6
votes
2answers
193 views
Example of a non-algebraic $\ell^2$-function in two variables
Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
-1
votes
1answer
243 views
inverse function theorem and surjectivity
I have read by the advice of Mr.mixed math,and Mr.willie wong that inverse of a multi variable function can be found out using the theorem present here ,so in that case the author mentions about ...
4
votes
1answer
94 views
Cutoff functions are not nice in their first derivative
This afternoon, while working out this answer by Nate Eldredge, I made some vain attempts at building cutoff functions of various kinds. Especially I was looking for the following.
Can a sequence ...
