Questions tagged [step-function]
A step function, also known as a simple function, is a finite sum of characteristic functions of bounded intervals. They are often used in measure theory to approximate integrable functions.
149
questions
0
votes
3answers
28 views
Integral of Heaviside($R^2-x^2-y^2$)
I have this integral (where $R$ is a positive constant) $$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} H(R^2-x^2-y^2)dxdy$$
and I'm pretty lost when trying to calculate it.
I don't ...
1
vote
1answer
35 views
Chain rule use in discontinuous generalized function derivative
Let's assume we have a function of the following form $f(x,a):=g(H(a-x))$, where $H$ is the Heaviside step function.
We now would like to look at the derivative $\frac{\partial}{\partial x}\int_0^1 f(...
1
vote
1answer
32 views
Computing the Fourier transform of three distributions - one last part
In my analysis class, we are now covering distribution theory. We take the Schwartz space of functions $S(\mathbb{R})$ and its continuous dual space, $S'(\mathbb{R})$, the space of tempered ...
0
votes
1answer
29 views
Riemann integrability of step functions
Suppose the step function is defined as follows. A function $f$ is a step function on $[a,b]$, if there exists a finite partition $P$ of $[a,b]$ such that $f$ is constant on the interior of each ...
0
votes
1answer
24 views
Length of integer repeats in an integer square root step function
Given the step function $f(n)=\lfloor\sqrt{n}\rfloor$ producing the integer sequence $$0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4,\ldots $$ I observed (conjecture) that the integers repeat an ...
0
votes
0answers
29 views
Construct a sequence of step function $f_k$ so that $f_k \rightarrow f$ a.e.
Let $f:[0,1]\times[0,1] \rightarrow \mathbb{R}$ be given with $f(x)=\frac{1}{||x||}$ for every $x≠(0,0)$.
Construct a sequence of step functions $(f_k)_{k\in\mathbb{N}}\in T^{inc}$, so that $f_k\...
0
votes
1answer
33 views
Approximate every monotonous increasing function f uniformly by a sequence $(f_n)_n$ of step functions, s.t. $f_{n+1}-f_n$ is monotonous increasing?
Is it possible to approximate every monotonous increasing function $f:[0,1]\rightarrow \mathbb{R}$ uniformly by a sequence $(f_n)_{n \in \mathbb{N}}$ of step functions on $[0,1]$, s.t. for all $n\in \...
0
votes
0answers
41 views
Expressing a function as a series involving Heaviside functions
Let $H(x)$ denote the Heaviside function
$$H(x)=\begin{cases} 0 \space \text{ if $x<0$}\\ 1 \space \text{ if $x>0$}\end{cases}$$
Suppose we want to express the following square function $g(x)$ ...
0
votes
1answer
27 views
Need help understanding how to set up the following problem dealing with step functions.
So with this problem I am completely lost on where to even start this problem to get to the answer that my professor wants me to prove. The questions is shown in the attached picture.
Let f(x) = abs(x)...
0
votes
1answer
32 views
Can every step function be partitioned into equally-sized subintervals?
I'd like to use the following fact as part of a proof on a problem set, but I'm not sure if it's true.
Let $P = \{x_1, x_2, ..., x_n\}$ be a partition of $[a, b]$, and suppose $s$ is constant on each ...
0
votes
1answer
19 views
Characterisation of null sets (sets of measure zero) as divergences of sequences of step functions.
In A. J. Weir’s Lebesgue Integration and Measure (CUP 1973) the author proves that, given an increasing sequence of step functions $\phi_n$ for which the sequence $\smallint \phi_n$ converges, the ...
0
votes
0answers
13 views
Does convolution of rectangles with different support converge to a gaussian?
Let $Y$ be the unit step function. If I convolve $Y * Y$ I get a triangle, convolve it again and the word "spline" starts to appear in my mind, and finally this $Y * Y * Y * ... * Y$ converges to a ...
0
votes
2answers
46 views
Convolution of x(t) and x(-t)
Consider the signal $x(t)=e^{-t}u(t)$ where $u(t)=\mathbb{1}(t\geq0)$, i.e. the Heaviside function.
Find the signal $y(t)=x(t)*x(-t)$
My attempt:
$y(t)=x(t)*x(-t)$
$=e^{-t}u(t)*e^{t}u(-t)$
$=e^{-...
1
vote
0answers
18 views
Fourier transform rule for $f(t)u(t)$ and Fourier transform over finite domain
I am looking for a rule to get the Fourier transform of $f(t)u(t)$ where $u(t)$ is Heaviside step function. In other words, assume I know $\mathcal Ff$ over the full real line $-\infty < t < \...
0
votes
0answers
51 views
Analytical Solution of Picewise-Non linear 4th order ODE
I'm strugling for a while on that. Is there a way to approach analytical the following BVP?
$$
\frac{d^4y}{dx^4}+[4\lambda_1^4H(y)+4\lambda_2^4H(-y)]y=0
$$
Where $H(x)$ stands for the Heaviside Step ...
0
votes
1answer
34 views
Does any such function exist given the following conditions
I would like to know if a step function $f$ exists on $\mathbb{R^+}$ such that for $k \in \mathbb{N}$,
$$f(0)=1$$
$$f(1)=0$$
$$f(2k) = (-1)^k$$
$$f(2k+1) = 0$$
Where $f(x) \not= \cos\bigl(\frac{\pi ...
2
votes
0answers
46 views
Proof that [(1 + √3) ²ⁿ⁺¹] is divisible by 2ⁿ⁺¹ ( [x] denotes the greatest integer function of x) for n >= 0
I came up with a proof but I am not sure if that is correct.
I am not sure whether this is rigorous proof, but I think I have a proof for the fact that $[(1 + \sqrt{3})^{2n+1}] = k2^{n+1}$ for $k \in ...
0
votes
1answer
38 views
Can you help on this question on Lebesgue Integrals
Suppose we have the measure space $[0,1]$ with the lebesgue measure, $\lambda$ on $[0,1]$. Define $g:[0,1] \rightarrow \mathbb{R}$ is measurable and that $g(x) > 0 $ almost everywhere on $[0,1]$. ...
0
votes
1answer
33 views
function $f:[0,1] \to \Bbb R$ s.t $f$ is not a step function, but for any $\epsilon \in (0,1)$ the restriction of $f$ to $[\epsilon,1]$ is.
I need to construct a function $f:[0,1] \to \Bbb R$ such that $f$ is not a step function, but for any $\epsilon \in (0,1)$ the restriction of $f$ to $[\epsilon,1]$ is a step function.
So my function ...
3
votes
2answers
55 views
Integral involving Heaviside function
For a class of Physics I need to compute the following integral:
$$\int_{-L}^{L}\mathrm{d}q\dfrac{\theta(\epsilon-bq)}{\sqrt{(\epsilon-bq)}}$$
and I truly have no idea on how to proceed. Note $\...
0
votes
0answers
28 views
Transfer Function of Electrical System
I am trying to find the transfer function $\frac{V_{out}(s)}{V_{in}(s)}$ of an electrical system that operates according to the diagram below.
$V_{in}$ is switched between 0 and 13 Volts (i.e $V_{in}(...
1
vote
0answers
17 views
Specific algorithm for approximation by step functions
I was wondering whether the following is a successful approximation in sup norm and is used. For a continuous function $f$ on an interval $[a,b]$, can be approximated by a step function $\phi$ defined ...
1
vote
1answer
53 views
Heaviside and trig function integral $\int \sin(3t)\theta(t)dt $
I can't figure out how you're supposed to find the solutions for a product involving trigonometric functions with heaviside, most examples online involve exponentials which have the nice property of ...
0
votes
0answers
16 views
Confusion regarding the unit step and unit impulse
So I understand that the unit impulse function is the "derivative" of the unit-step, but why is it that unit impulse functions are used to begin with?
My differential equations textbook describes it ...
1
vote
1answer
35 views
Finding the Laplace transform of a step function
I am trying to find the Laplace transform of the function $f(t)=(7-t)(u(t-1)-u(t-4))$ for $s \neq 0$. As far as I know this is a function of the form $f(t-c)u_c(t)$ where $u_c(t)=(t-c)$. As such I ...
3
votes
2answers
154 views
If $a \lt b$ and $f,g \in R[a,b]$ satisfy $f \leq g$ then $\int^b_af(x)dx \leq \int^b_ag(x)dx$
I am asked to prove the comparison property for regulated functions, namely :
If $a \lt b$ and $f,g \in R[a,b]$ satisfy $f \leq g$ then $\int^b_af(x)dx \leq \int^b_ag(x)dx$.
The definition for ...
0
votes
1answer
83 views
Approximating $1/x$ with a step function
I have a task in my textbook, which is the following:
For $f: [1,2] \rightarrow \Bbb R$ , $f(x)= 1/x$ choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n= \{r/n:n \leq ...
0
votes
1answer
70 views
Proving that sin$(1/x)$ is not regulated
I have seen in this question howto prove whether sin$(1/x)$ is not regulated. But i'm not quite sure why it's correct. Since the definition of a regulated function is as follows:
This means that the ...
1
vote
2answers
64 views
Find a reduction for $(H(t+3)-H(t-5))\cdot(\delta(t+2)+\delta(t-3)+\delta(t-9))$.
This exercise is from a Complex Analysis course, more explicitly inside the "Laplce Transform" chapter:
Find a reduction for $$(H(t+3)-H(t-5))\cdot(\delta(t+2)+\delta(t-3)+\delta(t-9)),$$ where $H(...
1
vote
0answers
17 views
Step-function Fit with Three Values for a Given Graph
Given a graph how can we find a step-function that can only take three values to best fit the graph?
For example, given this graph, where the function we want to fit is the one in black. What is the ...
0
votes
0answers
41 views
A Question on Step Functions and Measure Theory
I am self teaching myself on step functions and measure theory. There was a question I am working on in one of my textbooks. The question is this: Assume $ \{ \phi_n \} $ is a sequence of step ...
2
votes
1answer
123 views
Show a step function is measurable
Suppose $X_n$ is a step function that converges pointwise to $X$, where $X: \Omega \rightarrow \mathbb{R} $ is a measurable function. How would I show that that $X_n$ is measurable with respect to $\...
1
vote
0answers
21 views
Step functions : $\phi \leq \psi$ implies $\phi_i \leq \psi_i$?
In this question he has stated that $\phi \leq \psi$ implies $\phi_i \leq \psi_i$ for $i \in [1,k]$
I don't think this is correct as there may be a value $\psi_i$ which is smaller than $\phi_i$ ? Or ...
1
vote
1answer
38 views
Step functions: Using $-|\phi| \leq \phi \leq |\phi|$ conclude $|\int^b_a \phi(x)dx| \leq \int^b_a|\phi(x)|dx$
Let $\phi :[a,b] \rightarrow \Bbb R$ be a step function.
Using $-|\phi| \leq \phi \leq |\phi|$ I need to conclude that $|\int^b_a \phi(x)dx| \leq \int^b_a|\phi(x)|dx$
I have no idea how to prove ...
3
votes
0answers
53 views
$\phi$ is a step function. Prove that $|\phi|$ is a step function
Let $\phi :[a,b] \rightarrow \Bbb R$ be a step function.
I have to prove that $|\phi|$ is a step function. Here's how I prove it:
Let $P$ be a partition $P=\{p_0,...,p_k\}$ on $[a,b]$, compatible ...
1
vote
0answers
49 views
Step functions: $\phi \leq \psi$ then $\int^b_a\phi(x)dx \leq \int^b_a\psi(x)dx$
Let $\phi,\psi :[a,b] \rightarrow \Bbb R$ be step functions.
I have to show that if $\phi \leq \psi$ then $\int^b_a\phi(x)dx \leq \int^b_a\psi(x)dx$
How I go with proving this:
Let $P$ be a ...
0
votes
2answers
34 views
Sequence of step functions satisfying $\frac{\int^1_0\phi (t)dt}{||\phi_n||_{\infty}}\rightarrow 0 \text{ as $n \rightarrow \infty$}$
Is there a sequence of step functions $\phi_n :[0,1] \rightarrow \Bbb R (n \in \Bbb N)$ that satisfy $\int^1_0 \phi_n(t)dt \gt 0$ and $$\frac{\int^1_0\phi_n (t)dt}{||\phi_n||_{\infty}}\rightarrow 0 \...
0
votes
1answer
35 views
Approximation by step function
In my analysis class, I have learned that for any compactly supported and Riemann integrable function $f$ (with support $[a,b]$) and for any $\epsilon>0$, there exists a step function $g$ such that ...
0
votes
1answer
114 views
Proof that you can approximate any continuous function using rectangles/step functions within a small error
Proof that rectangles or a combination of step functions can approximate any continuous function within a small $\epsilon$ which represents the error between the approximate step function and ...
0
votes
1answer
26 views
Suppose a spring-mass system satisfies the inital value problem
$y''+0.25y=k[u_{1.5}(t)-u_{2.5}(t)]$
$y(0)=0$
$y'(0)=0$
where $k$ is a positive parameter.
a) Solve the initial value problem in terms of k.
b) Plot the solution for $k=1/2$, $k=1$, and $k=2$.
For ...
1
vote
1answer
73 views
Product of Heaviside distributions
In "A Smooth Introduction to the Wavefront Set", the product of two distributions is defined as follows.
Let $u, v \in D'(\mathbb R^n)$. We say that $w \in D'(\mathbb R^n)$ is the product of $u$ ...
0
votes
0answers
33 views
Step function and unit sphere, who's no a sequentially compact space
I've been asked following question.
The closed unit sphere is not a sequentially compact space. We define the closed unit sphere as $K := \{ f\in T[a,b] \mid \| f \|_\infty \leq 1 \}$
whereas $T[a,...
-1
votes
2answers
28 views
What is the equation for this graphed function? [closed]
I can't seem to figure out the equation for this simple function, I know that the equation for a vertical line is x = k but this looks like a kind of step function that is infinity when x = k.
0
votes
0answers
42 views
Steps to solve a Stieltjes integral with a step function?
I am trying to understand the steps for solving the equation:
$$\epsilon(t)=\int_0^tJ(t,t')\mathrm{d}\sigma(t')$$
assuming $σ(t)=H(t-t')$, which I understand is known as a Heaviside step function.
A ...
0
votes
0answers
102 views
Numerical approximations by staircase functions
Let’s say I want a numerical approximation to a function $f: [a, b] \rightarrow \mathbb{R}$. Most books on Numerical Analysis teach to approximate $f$ by a polynomial function, and they rarely use ...
-1
votes
1answer
27 views
Step Function Proofs [closed]
My notes give a lemma but I,m snuggling how the lemma actually works. Any help would be appreciated
3
votes
1answer
83 views
Is $\ a + b$ defined by $\ (a + b)(x) = a(x) + b(x) $ a step function where $\ a $ and $\ b $ are step functions
Note: x ∈ X$ is a step function where $\ X ⊂ R^n$ is a finite union of boxes and $\ a, b : X → [0,∞)$ are step functions. But I have only just started looking at step functions and I am struggling to ...
0
votes
0answers
67 views
Is this a valid proof that this function is integrable?
The question states:
Define the function (where $n$ is in the positive integers)
$$f(x)=\begin{cases}
x, & \text{if $x=\frac{1}{n^2}$} \\
0, & \text{if $x\neq\frac{1}{n^2}$}
\end{cases}$$...
0
votes
1answer
43 views
Show that $f(x)=x²$ is a Regulated function
Show that $f(x)=x^2$, $x\in[0,1]$ is a Regulated function by giving a Step function $g_n$ that converges uniformly to $f$.
Show that $\|g_n-f\| \to 0$
1
vote
0answers
37 views
Linear combination of step function
I am supposed to express the following function as a linear combination of step function: $\left | \varphi \right |,\varphi ^{3},\varphi -\gamma , max\left \{ \varphi ,\gamma \right \}$, where:
$$\...
