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.

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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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\...
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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 \...
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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 $g(x)$ ...
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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)...
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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 ...
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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 ...
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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 ...
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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^{-...
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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 < \...
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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 ...
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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 ...
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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 ...
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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 $[0,1]$. ...
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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 ...
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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 $\...
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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}(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $P_n= \{r/n:n \leq ...
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Proving that sin$(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 ...
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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(...
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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 ...
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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 ...
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1answer
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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 $\...
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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 ...
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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 ...
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$\phi$ 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 ...
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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 ...
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2answers
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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 \...
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1answer
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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 ...
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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 ...
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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 ...
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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 $u$ ...
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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,...
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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.
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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 ...
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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 ...
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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
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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 ...
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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) $$f(x)=\begin{cases} x, & \text{if $x=\frac{1}{n^2}$} \\ 0, & \text{if $x\neq\frac{1}{n^2}$} \end{cases}$$...
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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 $\|g_n-f\| \to 0$
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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: $$\...