Questions tagged [step-function]
The step-function tag has no usage guidance.
82
questions
1
vote
0answers
21 views
Sequences of step functions and measurable functions
I’ve found this result in my Measure Theory teacher’s notes.
If $f: \Omega \longrightarrow [-\infty, \infty]$ is measurable, then there exists an increasing sequence of step functions $f_n: \...
2
votes
1answer
24 views
Would the inverse Laplace transform of $\frac{e^{-s}}{s(s+a)}$ be $u(t-1)e^{-at}$?
I'm stuck on a problem and I'm not sure if I'm approaching it correctly. I'f I'm trying to find the Inverse Laplace Transform of:
$$f(s) = \frac{e^{-s}}{s(s+a)}$$
I would think that the answer would ...
0
votes
1answer
29 views
Is the sum of positive jumps from a jump process adapted again?
Let $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in [0;T]},\mathbb P)$ be a filtered probability space satisfying the usual conditions and let $\{X_t\}_{t\in[0;T]}$ be an adapted stochastic process, whose ...
0
votes
4answers
42 views
Solving O.D.E and Initial Values Problem using Laplace Transform
I have this ODE:
$$
y'' + y =
\begin{cases}
\cos t, &\text{ if }0\le t \lt \pi\\
t-\pi,&\text{ if }\pi \le t \lt \infty
\end{cases}
$$
The initial values are:
$$
y(0)=0 \\
y'(0)=0
$$
I ...
0
votes
1answer
39 views
Find Laplace transform of $t-\pi$
I am dealing with an Initial Value Problem of a step function:
$$
y'' + y =
\begin{cases}
\cos t, &\text{ if }0\le t \lt \pi\\
t-\pi,&\text{ if }\pi \le t \lt \infty
\end{cases}
$$
I am ...
0
votes
1answer
15 views
Step Function confusion
Suppose $g(x) = 1$ if $x=0$ and $g(x)=0$ otherwise, would that be a considered a step function? I assume it is because $g(x)$ can be written as the indicator function acting on $x=1$?
Sorry if this ...
0
votes
2answers
22 views
How to convert this piece-wise function to unit step? [closed]
How would I approach this piece-wise function to convert it to unit step?
$$g(t) = \begin{cases}2t & 0 \leq t<1\\2 & 1\leq t < ∞\end{cases} $$
How do I start this? I'm not getting the ...
-2
votes
1answer
32 views
Step function example
Give an example of a step function $s:[-1,3]\rightarrow \mathbb{R}$ such that $s([-1,3])$ contains at least $4$ distinct real numbers and
$\int_{-1}^3 s = \pi\,.$
$s(x)=-20 \text{ if } -1 \leq x &...
0
votes
1answer
11 views
Why must all co-efficents $c_j$ be positive in the integration of step functions?
I am working through a proof of the following theorem:
'If $f$ and $g$ are step functions having $f(x) \geq g(x)$ for all real values $x$, then $\int f \geq \int g$.
So far I understand and thus ...
0
votes
1answer
45 views
Properties of Heaviside Function
Let $H(x)$ be the Heaviside function defined by
\begin{cases}
1 & \text{if } x\geq0\\
0 & \text{if } x<0
\end{cases}
I know that
$H'(x)=\delta(x)$. The derivative of the Heaviside ...
1
vote
0answers
49 views
Approximation of Step function?
Is it possible to approximate the step function?
f(x)=\begin{cases}
0 \quad\text{ if }x\le 0\\
1\quad \text{otherwise}
\end{cases}
I want to implement it by polynomial approximation.
1
vote
1answer
50 views
How does a collection of step functions form a linear space?
Reading through a text book and it states that the collection of step functions form a linear space.
I know a step function has the form:
$$f(x) = \sum_{i=1}^{n}c_i \cdot m(I_i)$$
But what exactly ...
0
votes
0answers
23 views
What does this piece-wise function look like?
So I have $f(t) = t$ for $0\le t<1$, and $f(t + 1) = f(t)$ for all $t \ge 0$, i.e., $f$ is a periodic
function with period $T = 1$. I am wondering what this function actually looks like. I know ...
0
votes
0answers
16 views
Extract Fourier Transform from discrete time signal
I have two discrete time signals and I have to firstly determine their function form before determining their Fourier transforms and compare them.
Here are the two signals
As you can see, one is the ...
1
vote
0answers
33 views
How do you determine a step function to approximate another function?
I have a question here in which I need to explicitly write down a sequence $f_n(x)$ that can approximate $e^x$.
From reading, I known that I need to pick a partition sequence of $x_k$ so that I can ...
1
vote
1answer
29 views
Graph Of Step Functions
If we are asked to graph the step function $[\sqrt{x}]$ for $0 \leq x \leq 10$, I have seen a solution, which is given below,
what I am having doubts about is the closed circle at the end when $x=10$,...
1
vote
1answer
188 views
Laplace Transform of $ te^{2t}$ using unit step function
I was wondering if I could get some help with this question:
Consider the function:
$$f(t) = \begin{cases} te^{2t} \quad \,0 \leq t < 3\\
0 \quad \, 3 \leq t
\end{cases}$$
(a) ...
0
votes
0answers
24 views
What is the rule of integration of heaviside step function
I am trying to calculate an exterior multipole moment for a disc in the xy-plane and part of the integral involve a Heaviside function, i.e.:
$$\int_{0}^{\infty}r^{l + 1}\Theta(R-r)dr \tag{1}$$
I ...
0
votes
0answers
21 views
Question on measurable functions being approximated by step functions
This question is based on Theorem 4.3 in Stein's book. It's trying to show that $f = \chi_{E}$, where $\chi_{E}$ is the characteristic function on a measurable set $E$, can be approximated by step ...
-1
votes
1answer
29 views
Integrating the composition of a Heaviside function with a smooth function
I am trying to find how to compute an integral of the form:
$\int_{R^n}{\Theta(g(x))f(x)\,dx}$, where $\Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is ...
0
votes
0answers
36 views
Laplace Transform of Functions with Infinite Discontinuities
I know it's possible (generally speaking) to take the Laplace transform of step functions with a finite amount of discontinuities, such as $f(t) = u_0(t)$, $f(t) = u_3(t)\sin(t)$, etc. where $u_x(t)$ ...
0
votes
1answer
35 views
Representing rounding algebraically [closed]
Is there a standard way to deal with rounding in algebra? For example:
y = x + round(x/2)
Would give 2 when x = [1, 3), 3 when x = [3, 5), etc. This of course ...
1
vote
0answers
21 views
Contour Integral over Heaviside Function
I am stuck on how to do a problem where I am deriving the Green's function for a nuclear scattering system. I am currently starting with the expression
$$G^+(\mathbf{0})=\frac{2m}{(2\pi)^2}\int d\...
0
votes
1answer
21 views
An exemple of integral of distributions
Need to solve this integral:
$$I=\int_{-1}^{1}dx(\lim_{\varepsilon\to 0^+}\frac{\varepsilon}{\varepsilon^2+x^2}f(x)+\pi\vartheta(x)\frac{df(x)}{dx}(x)) $$
I think I should recognize the limit as a ...
0
votes
1answer
31 views
showing that $\|f\|_1=\sup \{\int_{[a,b]}\tau(x)dx \mid \tau \text{ step function and } \tau\le f\} $
Let $a,b\in\mathbb{R}$ such that $a<b$ and $f\colon [a,b]\to \mathbb{R}$ a non-negative function. Is then
$$\|f\|_1=\sup \{\int_{[a,b]}\tau(x)dx \mid \tau \text{ step function and } \tau\le f\} ?$$
...
1
vote
2answers
87 views
How do we calculate integrals without knowing differentiation?
To calculate the integrals, we use the fundamental theorem of calculus.
i.e for example to find $\int f(x)dx$, what we do is we find a function $g(x)$ such that $g'(x) = f(x)$. But, is there any way ...
1
vote
1answer
50 views
Why is $ \prod_{n=0}^{N-1} u[x_n + \theta] - u[x_n-\theta] = u[\theta - \max(|x_n| )]$?
I'm self-studying math, and came across a problem:
$$
\prod_{n=0}^{N-1} \left(u[x_n + \theta] - u[x_n-\theta]\right) = u[\theta - \max(|x_n| )]
$$
where $u$ is a unit step function, $x_n$ are sample ...
1
vote
0answers
183 views
Show step functions are Riemann integrable
We have $f: [a,b] \to \mathbb{R}$ is a step function if there exists a partition $P=\{x_0, \ldots, x_n \}$ of $[a,b]$ such that $f$ is constant on the interval $[x_i, x_{i+1})$.
I want to show that ...
1
vote
0answers
28 views
Heaviside step function with function as argument
Is the following computations correct?
Can the Heaviside step function have an arbitrary function as argument?
It seams reasonable and leads to the correct/same answers, but I have not been able to ...
1
vote
0answers
343 views
Fourier series of Heaviside step function?
Let us say we have the Heaviside unit step function $\Theta(t-t^\prime)$. I want to calculate its Fourier series
$$
\Theta(t-t^\prime)=\frac{1}{T}\sum_{n,m}\Theta_{\omega_n,\omega_m}e^{-i\omega_n t}e^{...
2
votes
1answer
121 views
Is the mentioned method appropriate to solve $\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$? [duplicate]
The integral is,
$$I=\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$$
I know the answer would be $\pi$ and I know how to solve this using Feynman's method and Fourier transform. However I was trying ...
0
votes
2answers
49 views
Continuity and Differentiability of Step function?
All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How ...
0
votes
1answer
35 views
construction of step functions to show integrability using Beppo Levi's theorem
Let $M\subset\mathbb{R}^n$ be measurable, $f\colon M\to\mathbb{R}$ continuous, bounded. Claim: $f$ is Lebesgue-integrable.
I was able to prove it for $M$ additionally bounded. How to reduce the ...
1
vote
0answers
32 views
Solving the integral of a step function
I am dealing with a step function S(t). The true functional form is not given or unknown, but what is known is that S(t) takes a different value at each time point t like this below.
...
0
votes
1answer
49 views
Heavyside function and Laplace transform
How to calculate the Laplace Transform of the following
$f(t)=-t^3u_3(t)+\cos{(3t)}u_6(t)$ ...(1)
Solution:- The Laplace Transform of $\cos{(3t)}u_6(t)$ can be calculated using $\cos{(at+b)}=\frac{...
2
votes
1answer
472 views
Convolution of $e^{-at}u(t)$ and $e^{at}u(-t)$
I have following convolution:
$$e^{-at}u(t)*e^{at}u(-t);a>0$$
$u(t)$ is the unit step function.
I have tried the following:
$$\begin{align*}e^{-at}u(t)*e^{at}u(-t)&=\int_{-\infty}^{\infty}e^...
0
votes
1answer
307 views
Is Heaviside step function or unit step function periodic?
I have a unit(or Heaviside) step function in discrete form:
$$\text u[n]=\begin{cases} 0, & n < 0, \\1, & n \ge 0, \end{cases}$$
and in continuous form:
$$\text u(t)=\begin{cases} 0, &...
0
votes
1answer
19 views
Examination whether a definition is well-defined
Let f be a positive measurable function and $\varphi_k$ a monotone sequence of positive step functions such that $\varphi_k \leq \varphi_{k+1}$ and $\varphi_k \rightarrow f$ pointwise. Then we define ...
0
votes
0answers
85 views
Fitting a curve to $N$ dimensional data
I have a data set with $N$ independent variables and one dependent variable(function of all the $N$ independent variables). The dependent variable is either $0$ or $1$ (like a step function). I am ...
0
votes
0answers
124 views
Derivative of Squared Step Function
while studying signals & systems I ended up in the following difficult situation after digging up the unit step and delta functions a lot (I wish I hadn't do so, but couldn't help :]). Here it is: ...
0
votes
2answers
48 views
Find Inverse Laplace of $\frac{e^{-2s}}{(s^{3})}$ and evaluate it $f(3)$.
I was able to solve it by using second shift theorem which led my answer to be $y(t)=u(t-4)-(t-4)^2/2$ but how would I evaluate it for $f(3)$?
I am unsure on what to do with u.
I tried to use unit ...
0
votes
0answers
55 views
Poisson point process with minimum event interval
A Poisson process is defined by the probability of encountering $k$ events within an arbitrary time interval $\Delta t$, namely
$P\{N(\Delta t) = k\} = \frac{e^{−λ\Delta t}(λ\Delta t)^k}{k!}$
...
1
vote
1answer
55 views
How to integrate greatest integer and fractional part
Integrate this function: $⌊x^2⌋ + \{x^2\}$ , from $0$ to $2$
I know that the first part is a step function, and every integer value, say, from 1 to 2, will be 1. But I can't understand how to ...
0
votes
0answers
84 views
Multivariable integral, use of Dirac Delta and Heaviside Theta
$\newcommand{\diff}{\operatorname{d}}$
$\newcommand{\deriv}[2]{\frac{\diff #1}{\diff #2}}$
$\newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}}$
$\renewcommand{\vec}[1]{\boldsymbol{#1}}$
I have ...
0
votes
0answers
33 views
Any way to prove this identity :$\frac{d^m}{dx^m}\frac{H(x)x^{m-1}}{(m-1)!} =\delta(x)$?
One of my friend sent me to proof this identity :$\frac{d^m}{dx^m}\frac{H(x)x^{m-1}}{(m-1)!} =\delta(x)$ Where $\delta$ is the dirac delta function and $H$ is heaviside step function , I knwo only ...
1
vote
1answer
56 views
Laplace transform of $e$ raised to Step $e^{u(t)t}?$ [closed]
How I can transform this?
$e^{u(t)t}$
where $u(t)$ is the step function.
with
domain
[0, +∞[
3
votes
2answers
246 views
Integrating Heaviside Step Function of two Variables
Suppose we have a definite integral like $$ I=\int_0^\infty dx \int_0^\infty dy \, Θ(α-x-y) $$
where $a \in R_+^*$ and $Θ$ is the Heaviside step function. Of course this is easy in that we can find ...
1
vote
1answer
80 views
Double integral with nested heaviside function
I encounter the following form of integrals and I would like some suggestions on how I can solve it analytically (numerically is straightforward):
$$ F(a,b,c) = \int_{0}^\beta \int_{0}^\beta \int_{0}...
0
votes
1answer
78 views
Expressing a step function
Take the interval $I=[0,1]$ and the sequence $\{x_n\}_{n\in\Bbb{N}}$ defined as follows:
$$x_n=\sum_{k=1}^n \frac{1}{2^k}, \forall n \in \Bbb{N}$$
and $x_0=0$.
Now, that sequence defines a ...
1
vote
1answer
40 views
How can $\int_0^x\lfloor t \rfloor^2dt$ be written as $\sum_{j=1}^{\lfloor x - 1 \rfloor} j^2 + q^2r$
Question 6(c) from Section 1.15 Exercises of Apostol's Calculus is the following:
Find all $x > 0$ for which $\int_0^x\lfloor t \rfloor^2dt = 2(x-1).$
A particular piece of reference material ...
