# Questions tagged [total-variation]

This tag is for questions relating to Total Variation.The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper (Jordan 1881). He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.

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### Total variation and probability measures

I'm currently reading some notes about optimal transport and here there is the definition of total variation:$$\Vert\mu\Vert_{TV}=2\sup\vert\mu(A)\vert$$where $\mu$ is a probability measure of the ...
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### Inversely proportional proof [closed]

Let $$y \propto \frac{1}{x}$$ Then we know y is inversely proportional to x. Then, Why is y is inversely proportional to x same as $y \propto \frac{1}{x}$. My attempt to prove this:- Let $k$ be the ...
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### Total variation distance on product space

Let $X$ be a measurable space. Given probability measures $p$ and $q$ on $X$, define their total variation distance as $$d(p,q) = \sup_f \Big| \int f \, dp - \int f \, dq \Big| ,$$ where $f$ varies ...
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### intuitive interpretation of total variation [duplicate]

Assuming a nicely-behaved function (continuous, differentiable). Total variation can be defined as $T_f=\int_a^b|f'(x)|dx$. My intuitive understanding of what total variation measures is: the length ...
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### Understanding total, quadratic, and $\Phi$ variation of functions

I've started to study stochastic calculus on my own recently (I'll read Fima's book for a simpler introduction and then Steele's for a maybe more formal approach). I've come across the definition of ...
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### question on total variation

I read this sentence THEOREM 5.62. A function $f : [a,b] → R$ is of bounded variation iff f can be written as the difference of two increasing functions , that is, f = g−h on [a,b] for two increasing ...
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### Bounding the total variation of an Absolute Continuous function

Theorem: Suppose $f \in AC([a,b])$. Prove that $$V_a^b(f) \leq ||(f')^+||_{L^1([a,b])} + ||(f')^-||_{L^1([a,b])}$$ Proof (my attempt): If $f \in AC([a,b])$ then calculus formula holds and we can ...
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### Fourier series of function with bounded variation is bounded by its norm

We have $\mathbb{T} = \mathbb{R}/\mathbb {Z}.$ We define a space $BV(\mathbb{T}) = \{f| f:\mathbb{T} \to \mathbb{C}, V(f) <\infty \},$ where $V(f)$ denotes the total variation. We also equip this ...
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### Computing the total variation for a multivariable function

I am trying to write an example computation with multivariable total variation to include in my functional analysis notes using the following definition from Wikipedia: Let $\Omega$ be an open subset ...
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### characterizations of singular measures

If $\nu$ and $\mu$ are finite and positive measures then $\nu \perp \mu$ if and only if $\|\nu - \mu \|= \|\nu\| + \|\mu \|$ Here, $\nu \perp \mu$ means that $\nu$ and $\mu$ are singular measures, ...
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### How Jacobian is defined for the function of a matrix?

Let $f: \mathbb{R}^{m\times n} \rightarrow \mathbb{R}^m$ where $f(W)=Wx$. Question1: How is the Jacobian matrix defined for a vector-valued function whose variable is a matrix? Question2: Using the ...
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### total variation, signed measure, inequality

Suppose that $\mu$ is a signed measure on a $\sigma$-algerba of the subsets of X. Show that: $\frac{1}{2}\lvert{\mu}\rvert(X) \leq \sup[{\vert{\mu(E)}\rvert: E \in \sigma-algebra}]$ also 1/2 is the ...
This is exercise 6.5 of the book Finite Volume Methods for Hyperbolic Problems by R.J. LeVeque (2002). Show that the minmod slope guarantees that $$TV(q^n(·, t_n)) ≤ TV(Q^n) \tag{6.23}$$ will be ...