3
votes
1answer
38 views

Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$

A function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(2)=2$ and $$f(x+1)+f(x-1)=\sqrt{3}f(x) \tag{1}.$$ Find the values of $f(0)$, $f(4)$, $f(6)$ and $f(18)$. My approach: replace $x$ ...
0
votes
0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
0
votes
0answers
92 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
0
votes
2answers
45 views

Matrix of Functions

I was thinking about a problem and I realized for a family of functions say $L^1([a,b])$ one can define matrices with functions as elements: $$ A=\left[ \begin{array}{ccc} f_{11}& ...
0
votes
1answer
25 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
1
vote
1answer
53 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
0
votes
0answers
50 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
3
votes
3answers
43 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
1
vote
0answers
28 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
0
votes
1answer
24 views

Dirac delta function and well behaved function [duplicate]

whether dirac delta function a well behaved function? Can u please explain the properties of a well behaved function..?
6
votes
1answer
156 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
1
vote
2answers
70 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
2
votes
0answers
43 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
0
votes
0answers
45 views

implicitly define a function

The first part i made $u=\frac{z}{x}$ and $v=\frac{y}{x}$ and after calculating the partial derivatives $\frac{dz}{dx}$ and $\frac{dz}{dy}$ The second i have no idea how to do it
0
votes
1answer
58 views

$f(tx,ty,tz)=t^n \cdot f(x,y,z)$ [closed]

For $t>0$ and $(x,y,z)\in \mathbb{R}^3$ show that if $f(tx,ty,tz)=t^n \cdot f(x,y,z)$ then $$x*\frac{\partial }{\partial x}f+y*\frac{\partial }{\partial y}f+z*\frac{\partial }{\partial z}f=n ...
0
votes
1answer
23 views

Definition of the total variation of a function $g:\mathbb{R}\to\mathbb{R}$

if the total variation of a a real function $f:[a,b]\to \mathbb{R}$ over $\textbf{P}=\{a=t_0<t_1<...<t_m=b\}$ is $$ V^{a}_{b}(f)=\sup_{\textbf{P}}V(f,\textbf{P}) $$ where $$ ...
0
votes
1answer
34 views

Can we approximate $f(x) = \chi_{(0,\infty)}(x)$ by smooth monotone functions?

Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence? Preferably ...
0
votes
0answers
25 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
0
votes
3answers
63 views

How to make a cos function into a sin function

I need to convert this equation into a sin function: f(x) = 12 cos(2x + 1) − 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem
1
vote
1answer
29 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
1
vote
2answers
20 views

Investigating a function with a parameter

I got stuck on solving this problem: For which $a \in \Bbb R$ is the function $$ f_a: \ ]1, \ \infty[ \; \longrightarrow \ \Bbb R: x\mapsto \frac{\log x}{(x-1)^a} $$ continuous on $[1, \ ...
0
votes
3answers
39 views

Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in ...
2
votes
0answers
38 views

In the space $L^2 [0,1]$ to solve for all values ​​of the complex parameters $\lambda$ and $b:$ [closed]

In the space $L^2 [0,1]$ to find a solution of the integral equation for all values ​​of the complex parameters $\lambda$ and $b$: $x (t)-λ\int_0^1 t^2s^2x(s) \, ds = 4t + bt^2$
0
votes
1answer
60 views

Prove that the shift operator $(Ax)(t)=x(t−a)$ not compact.

Prove that the shift operator $(Ax)(t)=x(t−a)$ not compact in the space of bounded continuous functions on R.
2
votes
1answer
38 views

Second derivative test inconclusive, all derivatives are 0, moving critical point to origin, no result?

Here is a function $f(x,y)=x^4 + 6x^2y^2 + y^4 -4x^3 - 12xy^2 + 6x^2 + 6y^2 - 4x + 1$. I've happily proved that $(1,0)$ is a critical point for that function. Now I'd like to decide whether is it a ...
0
votes
0answers
22 views

Prove and find the norm of investments $J: \ell_p \to \ell_q$ [duplicate]

Prove and find the norm of investments $J: \ell_p \to \ell_q$, $1 \leq p < q \leq\infty$.
0
votes
1answer
24 views

Contraction map

I have a general question about the properties of contractive/non-contractive maps. Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some ...
1
vote
1answer
55 views

prove that using uniform bounded theorem [duplicate]

Let $y=(\eta_j),\eta_j\in \mathbb C$, be such that $\sum \xi_j\eta_j$ converges for every $x=(\xi_j)\in c_0$ where $c_0\subset l^\infty$ is the subspace of all complex sequences converging to zero. ...
1
vote
1answer
37 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
1
vote
1answer
41 views

continuous extension and smooth extension of a function

Let $X$ be a metric space. Let $E$ be a subset of $X$. (1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
1
vote
1answer
54 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
0
votes
0answers
24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
1
vote
2answers
19 views

Is this function well defined for any $g\in L^q(\Omega)$?

If $p,q\in(1,\infty)$ such that $\frac 1q+\frac1p=1$, given $g\in L^q(\Omega)$ we difine: $$\Phi(g):L^p(\Omega)\to\Bbb R \\ \Phi(g)(f):= \int_{\Omega}fg$$ I know this is a basic question, but how do ...
0
votes
1answer
20 views

Periodicity and period of a function

The question is : Let $f(x)$ be a real valued function defined for all real numbers x such that for for some fixed real number $a>0$, $f(x+a)=\frac{1}{2} + \sqrt{f(x)-(f(x))^2}$ and $\frac12\le ...
0
votes
1answer
16 views

Establishing convexity of a function

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and ...
0
votes
1answer
39 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
0
votes
1answer
42 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
0
votes
1answer
32 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
0
votes
1answer
53 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
0
votes
1answer
44 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
3
votes
1answer
57 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
2
votes
2answers
73 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
0
votes
0answers
24 views

A function defined by a differential rule

I have read this: Define the non-decreasing function $Z(s)$ defined by the differential rule $$dZ=\min(ds, d\Phi(s))$$ and define its approximations $$Z_n(s) = \int_0^s \min(1, ...
3
votes
2answers
48 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
0
votes
0answers
15 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
2
votes
1answer
41 views

Examples of contractions between functional spaces

Define $\mathcal{F}$ as the following set of continuous functions: $$ \mathcal{F} := \left\{ f: \mathbb{R} \rightarrow \mathbb{R}^n \mid f(\cdot) \ \text{contin.}, \ f(x) \in K(x) \subset ...
1
vote
0answers
31 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
0
votes
1answer
71 views

Definite integral involving bessel functions of first and second kind

Is there any standard solution of the integral: $\lim_{\epsilon \to 0} \int_{\epsilon}^{a} J_m(k_1\rho)Y_m(k_2\rho)\rho \, d\rho$. where the integer $m\geq0$ and $a<\infty$
0
votes
1answer
60 views

Definite integral of product of two bessel functions of different order and different argument

What is the solution of the integral: $\int_0^a J_m(k_2\rho)J_{m+1}(k_1\rho)d\rho$ where the integer $m\geq0$
1
vote
1answer
37 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...