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 real analysis and measure theory to approximate integrable functions.
259 questions
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Determine if $\sum_{n = - \infty} ^ {\infty} e^{-(2t -n)}u(2t -n)$ is periodic. If so, what is its period?
I am given a continuous function $x(t) = \sum_{n = - \infty} ^ {\infty} e^{-(2t -n)}u(2t -n)$, where $u(t)$ is the unit step function, and asked to find its period, if it exists. I'm not sure how to ...
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A compact function for generating aperiodic step functions: precedents or applications?
In a recent paper in statistical physics, I used a function I constructed to represent an arbitrary $1$D configuration of discrete spin states (+1/-1) in a continuous limit. I found its form to be ...
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Terminology for piecewise linear function interpolating between constant 0 and constant 1 [duplicate]
Had some trouble writing a succinct and meaningful title here, so feel free to improve it. Category tags are also not great but I didn't see anything (other than "terminology") which was a ...
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Set of step functions on $[a,b]$ is dense in the set of right-continuous piecewise continuous functions
Let $S[a,b]$ be the set of all step functions and $PC[a,b]$ the set of all bounded piecewise continuous functions on $[a,b]$ which are continuous from the right and for which the limit at the left ...
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Where's my mistake in proving $\int_a^b f(x)\space dx = \frac{1}{k} \int_{ka}^{kb} {f(\frac {x}{k}) \space dx}$ for $k<0$?
I'm trying to prove that for $k<0$,
$$\int_a^b f(x)\space dx = \frac{1}{k} \int_{ka}^{kb} {f(\frac {x}{k}) \space dx}$$
In the book Calculus I by Tom Apostol, he proved this for $k>0$ and left ...
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Integral of an unusual "weighting" function
In a lecture yesterday on the definition of integrals in terms of step functions, my lecturer mentioned an unusual function on the interval [0,1]. I am curious to wether my evaluation of the integral ...
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Reconstructing the Riemann Integral Using the Completion of Normed Spaces
Every normed space can be completed into a Banach space, and this completion satisfies a universal property:
Let $X$ be a normed space, $\tilde{X}$ its completion, and $c: X \to \tilde{X}$ the ...
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Doubt in the derivation of DTFT of step function $u[n]$?
I have seen many answers to this question saying "Split it into even and odd parts and find corresponding transforms". In particular, for the odd part (say $u_{odd}[n]$), many used $u_{odd}[...
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Sorting a measurable function
Given a measurable function $f:[0, 1] \rightarrow \mathbb{R}$, I would like to "sort" (almost all) its values from least to greatest. For example, the "sorted" version of $f(x) = \...
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How does this definition of integral for bounded functions derive the properties of integral for step regions?
In the book Calculus I by Tom Apostol, he defined the integral of a bounded function as such,
Let $f$ be a function defined and bounded $[a,b]$. Let $s$ and $t$ denote arbitrary step functions ...
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How to emphasis on value $c$ in the definition of unit step function
Let $c\geq 0$. The unit step function $U(x-c)$ is defined by
$$
U(x-c)=
\begin{cases}0,\quad 0\leq x < c, \\
1,\quad x\geq c.
\end{cases}
$$
As I didn't find which one of the followings can be used,...
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Solve $\ddot x + 2 \dot x = H(t)$ where $H$ is the Heaviside unit step function
Find the unit step response of the system $$\ddot x + 2 \dot x.$$ That is, find $x$ in the ODE $$\ddot x + 2 \dot x = H(t)$$ where $H$ is the Heaviside unit step function
If this is impossible, or ...
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Equation in the book Recherches analytiques sur la théorie des nombres premiers by Charles-Jean de La Vallée Poussin
In the book Recherches analytiques sur la théorie des nombres premiers by Charles-Jean de La Vallée Poussin, in page 53, he writes:
$$\int_{1}^{y}dy\left[\frac{\sin(\beta\log y)}{y}\sum_{p<y}\log p\...
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Difference between Fourier transform of $e^{-at}u(t)$ using basic definition and frequency shifting property
Hopefully this a short question.
$u(t)$ is the Unit step function or the Heaviside function. $a$>0
Solving using the basic definition,
$$F[x(t)]=\int_{-\infty}^{\infty}e^{-at}u(t)e^{-iwt}$$
we get
$...
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Integral of a function by bounding between two step-functions
Take two step-functions $t(x)$ and $s(x)$ such that non-negative monotic $f(x)$ lies between them. $s(x) \leq f(x) \leq t(x)$. Then,
$$\int_a^b s(x)\leq I \leq \int_a^b t(x) \implies a(S) \leq I \leq ...
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