Questions tagged [step-function]

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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: \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 $x=10$,...
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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) ...
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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 ...
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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 ...
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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 ...
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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 $u_x(t)$ ...
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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 ...
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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\...
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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 ...
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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 $$\|f\|_1=\sup \{\int_{[a,b]}\tau(x)dx \mid \tau \text{ step function and } \tau\le f\} ?$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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^{...
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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 ...
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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 ...
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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 ...
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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. ...
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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{...
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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^...
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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, &...
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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 ...
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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 ...
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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: ...
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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 ...
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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 $P\{N(\Delta t) = k\} = \frac{e^{−λ\Delta t}(λ\Delta t)^k}{k!}$ ...
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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 ...
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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 ...
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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 ...
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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, +∞[
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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 ...
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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}...
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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 ...
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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 ...