# Questions tagged [step-function]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...