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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26 views

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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36 views

It is true for a continuous differentiable bounded function $f(x)$ that its total variation $V_a^b(f)\ge\max_{a<x<b}f(x)-\min_{a<x<b}f(x)$?

I am trying to understand the intuition about the total variation: at first I believe it leads to the length of the codomain/range of the function $f(x)$, which I think is equivalent to $\Delta_y(f) = ...
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36 views

Real Analysis: If $[a',b']$ is a subinterval of $[a,b]$ show that $P[a',b'] \leq P[a,b]$

If $[a',b']$ is a subinterval of $[a,b]$ show that $P[a',b'] \leq P[a,b]$ where $P$ is the positive variation defined by $P=sup \sum_{i=1}^m [f(x_i)-f(x_{i-1})]^+$ where $x^+$ is defined by $x$ if $x&...
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1answer
22 views

Mollification of Functions of bounded variation

Let $\eta_{\epsilon}$ be a standard mollifier. For $f\in L^{\infty}(\mathbb{R})$ we have $|f^{\epsilon}(x)|\leq \int\limits_{R}|f(x-y)|\eta_{\epsilon}(y)dy \leq ||f||_{\infty},$ which implies $||f^{\...
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1answer
62 views

Properties of functions of bounded variation

Consider the following definitions of the functions of bounded variation. Defintion 1. $f$ in said to be a function of bounded variation if $f\in L_{loc}^1(\mathbb{R})$ and $\sup\left\{\int\limits_{\...
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37 views

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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20 views

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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26 views

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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111 views

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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63 views

Compact subset of infinite-dimensional space has empty interior

My question is related to this question. My space is the set of all Borel probability measures on $\Theta=[0,1]$, which is a compact metric space under the Prokhorov metric. Call this space $\Delta \...
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29 views

Variations of a function defined by an integral

Given $f \colon [a,b] \to \mathbb{R}$ and $\mathcal{P} \colon a = x_0 < \cdots < x_m = b$ a partition of $[a,b]$. We define three possible variations for $f$: $P_a^b(f) = \sup\{p(f, \mathcal{P})...
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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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30 views

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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71 views

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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289 views

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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33 views

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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40 views

Total variation distance between binomial distributions

I would like to show that $$d_{TV}\left(\mathrm{Binom}\left(k,\frac{1}{2}\right),\mathrm{Binom}\left( k-k^\beta,\frac{1}{2})+k^\beta \right)\right)\to \begin{cases} 0 &\quad\text{if }\...
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1answer
55 views

How to show boundedness of a sequence in $BV$

For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e.. I'm ...
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19 views

Weighted total variation

Suppose $ P = \{x_1, x_2, \dots, x_n \} $ is a partition of $ [0, 1] $, i.e., $0 = x_1 < x_2 < \cdots < x_n = 1 $. Suppose $ g $ and $ w \geq 0 $ are smooth functions on $ [0, 1] $. I'd like ...
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42 views

Total variation does not take into account large distances.

Let $(X,\Sigma,\mu)$ be a measure space, then define $$|\mu |(E):=\sup\limits_{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma $$ where the supremum is taken over all partitions $\pi$ of the ...
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Comparison of definitions for Functions of Bounded Variation

I have been trying to understand the functions of bounded variation and I came across the following definitions Defintion 1: A function $f:\mathbb{R^d} \rightarrow \mathbb{R}$ is of bounded variation ...
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When does $p$-variation mesh size tend to $0$?

Let $u: [a,b] \to \mathbb{R}$ and define its $p$-variation as $$V_p(u) := \sup \left\{ \left(\sum_i |u(x_i)-u(x_{i-1})|^p\right)^{1/p}\right\}, $$ where the supremum ranges over finite partitions $P=\{...
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32 views

Total Differential Exercise: Production Function in Growth Rates.

I'm struggling to understand how Khan & Reinhart (1990) go from the next production function. $$y=A f(K,L,Z)$$ Where $y$ is the production of the economy, $A$ is a variable which contains the ...
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6 views

Bounding angle between PMF vectors and their total variation

Let $a=(a_1,\dots,a_N)$ and $b=(b_1,\dots,b_N)$ be two probability mass functions that map universe $[N]:=\{1,\dots,N\}$ to probabilities where $\sum_{i\in [N]} a_i = 1, \sum_{i\in[N]} b_i = 1$. Can ...
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50 views

Alternative characterization of total variation of $L^1$ functions

This is about exercise 3.3 from Ambrosio, Fusco and Pallara Functions of bounded variation and free discontinuity problems. I struggled with this for some time. Let $\Omega$ be an open set and $u \in ...
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30 views

Is total variation distance always achieved?

Given a measurable space ($\Omega$,$\mathcal{F}$) and two probability measures $\mu$, $\nu$. We define total variation distance as $$\lVert \mu - \nu \rVert_{TV} = \sup_{A\in \mathcal{F}}|\mu(A)-\nu(A)...
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1answer
109 views

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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48 views

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 ...
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1answer
76 views

minmod slope total variation diminishing

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 ...
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71 views

LeVeque excercise Total Variation

In Randall LeVeque Finite Volume Methods I've come across the Exercise 6.2 and 6.4. 6.2 Compute the total variation of a). $$q(x)=\left\{\begin{array}{ll}1&\text{if }x<0\\\sin(\pi x)&\text{...
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111 views

Bounded Variation and Weak Derivatives.

The total variation of a function $v\in W^{1,1}(\mathbb{R}^d)$ is equal to the $L^1(\mathbb{R}^d)$ norm of its weak derivative, see Total variation of (weakly) differentiable functions (an actual ...
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38 views

Properties of functions of bounded variation in multi-dimension

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a function of bounded variation. Then do we have \begin{eqnarray} \int\limits_{\mathbb{R}^{2n}}|f(x)-f(y)|\phi_{\epsilon}(|x-y|)dx dy \leq \epsilon TV(f) ...
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48 views

Arc length and total variation of a Lipschitz function

I am asked to show that if $f$ is a Lipschitz function defined in $[a,b]$, then the arc length of it's graph must be grater than the total variation of the function. I know that the arc length of a ...
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28 views

Inequality about complex measure

While I read Folland Real Analysis, I have some confusion about the total variation of complex measures. He defines the total variation as following Also I have inequality $|\nu(E)| \leq |\nu|(E)$ ...
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53 views

total variation of function equals to sum of total variation restricted to subintervals

Define $V(f,[a,b]) = sup \{\sum\limits_{i = 1}^n |f(x_i) - f(x_{i-1})| : a \leq x_0 \leq x_1 ... \leq x_n = b\}$. I am not assuming $V(f,[a,b])$ to be finite. I think that $V(f,[a,b])= V(f,[a,c]) + V(...
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Why are we interested in total variation and how does it tie into its integral definition?

I am studying measure theory and had a few questions. In the text I am using, it defines total variation of a measure $\nu$ as $$|\nu| = \nu^+ + \nu^-$$ My first question is why are we interested in ...
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1answer
145 views

Optimal transport and total variation distance

I have a question regarding the following concept equating total variation distance with a particular case of optimal transport. I don't understand why equality (6.11) holds. We know by Kantorovich ...
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17 views

Variational method and the weighting function

Why the variational function, while writing the weak form, is called the weighting function? Is there any specific region behind it?
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84 views

Total Variation Between Finite Measures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}^k$ be distinct and $0\leq a_i,b_i\leq 1$ be such that $\sum_{i=1}^n a_i = 1 = \sum_{i=1}^n b_i$. Define the finite measures $\mu=\sum_{i=1}^n a_i \...
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90 views

Tightness of subadditivity of Total Variation Distance (on Product Measures)

As can be observed in this post we have that total variation distance between 2 independent (discrete) product measures has a subadditivity property with respect to total variation of the marginal ...
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1answer
31 views

Total functions [closed]

Let $A = \{0,1,2\}$. Determine all total functions $f:A \rightarrow A$ for which $f^2(x) = f(x)$, where $f^2 = f \circ f$ and find how many such total functions are there? My thoughts: I know what is ...
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1answer
288 views

Total variation of almost all Brownian motion paths is infinite. Some doubts along the proof

I quote Schilling, Partzsch (2012). Let $(B_t)_{t\ge0}$ be a one-dimensional Brownian motion and $(\Pi_n)_{n\ge 1}$ be any sequence of finite partitions of $[0,t]$ satisfying $\lim\limits_{n\to\infty}...
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1answer
56 views

Definition of total variation: I cannot grasp its meaning

Starting from this definition of Total Variation for a function of one real variable, I cannot understand what it does mean that "supremum runs over set of all partitions". My ...
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1answer
27 views

How can we use the partial derivative in such a case?

Suppose that $g:(0,+\infty)\times(0,+\infty)\rightarrow \mathbb{R}$ is a function of $x$ and $y(x)$, where $x\in (0,+\infty)$ and $y:(0,+\infty)\rightarrow (0,+\infty)$ and y is a linear and ...
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2answers
85 views

Show $|\int f d \nu| \leq \int |f| d |\nu|$

Let $\nu$ be a complex measure and $f \in L^1( \nu)$. Prove that $$\left|\int f d \nu\right| \leq \int |f| d |\nu|$$ Here $|\nu|$ is the total variation of $\nu$. I managed to prove that the result ...
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2answers
123 views

Show that if $\nu(E) = \int_E f d \mu$, then $|\nu|(E) = \int_E |f| d\mu$

Consider the following fragment from Folland's book on real analysis: I'm trying to show that if $$\forall E \in \mathcal{M}: \nu(E) = \int_E f d \mu \quad \quad(d \nu = f d \mu)$$ where $f \in L^1(\...
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1answer
133 views

Pinsker's Inequality for Metric Spaces!

Let $p$ and $q$ be two probability distributions on a countable set $X$. Then the total variation distance $V(p,q)$ between $p$ and $q$ is defined as follows: \begin{equation} V(p,q)=\frac{1}{2}\sum_{...
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1answer
77 views

Folland complex measures total variation definition

Consider the following fragments from Folland's text on real analysis: Why can proposition 3.9 be applied to show uniqueness? I.e. why is $\rho = \mu_1 + \mu_2$ $\sigma$-finite?
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68 views

Complex-valued Martingale with Square Integrable Variation on Finite Intervals

Let $(X_t)_{t \geq 0}$ be an $\mathbb{R}^d$-valued compound Poisson process defined on a probability space ($\Omega, \mathbb{F}, P)$. Define the centered, complex-valued martingale $(M_t)_{t \geq 0}$ ...
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1answer
37 views

Complex vs. Real Total-variation Norms of Real Radon Measures

Let $\mu$ be a real Radon measure over $\mathbb{R}$. Its total variation norm is given by \begin{equation} \lVert \mu \rVert_{\mathcal{M}(\mathbb{R}, \mathbb{R})} = \sup_{f \in \mathcal{C}_0(\mathbb{R}...