Questions tagged [bounded-variation]

For questions about functions $f$ defined on an interval $[a,b]$ such that there exists a constant $M>0$, such that if $a=x_0<x_1<\ldots<x_n=b$, $n\in\mathbb N^*$, then we have $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq M$. This concept can be generalized to infinite intervals, requiring that the constant is uniform.

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Convergence in Total Variation vs implies Uniform Convergence

Let $(f_n)_{n\in \mathbb{N}}$ and $f$ real function defined on an interval $[a,b]$ s.t. $f, f_n$ are bounded variation function on $I$, and suppose that $TV(f_n) \to TV(f)$. I should prove that it ...
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Function of bounded variation in terms of a sequence

Let $(a_n)$ be a sequence of positive numbers and define $$f(x) = \begin{cases}a_n, \text{ if } x = 1/n, \ \ n \geq 1\\ 0, \text{otherwise}\end{cases}.$$Show that $f$ is of bounded variation on $[0,1]$...
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Equality of continuous functions of bounded and unbounded variations [closed]

Suppose $f:[0,1] \to \mathbb{R}$ is continuous and has infinite total variation on any interval $[a,b] \subset [0,1]$ with $a <b$. Suppose $g$ is a function of bounded variation on $[0,1]$ and is ...
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Suppose $f:[0,1]\to\mathbb R$ is continuous, possibly with unbounded variation. We consider sums of the form $$\sum_{i=1}^n\Big(f(x_i)-f(x_{i-1})\Big)^3$$ where $0=x_0<x_1<x_2<\cdots<x_{n-... • 5,434 5 votes 2 answers 278 views Bounding$L^1$norm of the difference between a function$f:\mathbb R^n\to\mathbb R$of bounded variation and a piecewise constant approximation As a follow up to this question, which deals with univariate functions, I assume that we are given a function$f:\mathbb R^n\to\mathbb R$which is of bounded variation on bounded sets, meaning, ... • 3,298 3 votes 1 answer 72 views If$f$is bounded variation then$|f|$is bounded variation I want to show that if$f$is of bounded variation in$[a, b]$then$|f|$is of bounded variation in$[a, b]$. Suppose that$f$is of bounded variation then if$P=\{x_{0}, \dots, x_{n} \}$is a ... • 1,730 1 vote 0 answers 44 views Bounded Variation In Compact set Let$D$be any finite collection interval on$E$and function$f:E\to\mathbb{R}$. If$E$is a compact set, show that $$V(f,E)=\sup\{V(f,D) \mid \text{for } D \text{ is any finite collection interval ... • 11 3 votes 1 answer 82 views Naturalness of definition of line integral Let I = [a,b] denote some interval, and f : I \rightarrow \mathbb{C} a continuous function of bounded variation, in other words, f is a parameterisation of \gamma = f(I), a rectifiable curve. ... • 1,622 2 votes 2 answers 126 views Bounded variation of functions and sequences I would like to know if there is a relation between bounded variation of functions and bounded variation in sequences. I know one can not directly correlate the concepts but if we consider the Fourier ... • 199 1 vote 0 answers 111 views How pathological can a function of bounded variation be? For f: [0,1] \to [0,1] , Define A_f = \{y : |\{f^{-1}(y)\}| > \aleph_0\}. It is known that A_f must have null measure if f is continuous and of bounded variation. This thread explains the ... • 11 6 votes 1 answer 182 views Approximating a function of bounded variation function by a stepfunction obtained by averaging over measurable sets Inspired by this question, suppose that we have a function f\in\mathrm{BV}([0,1])\cap L^1([0,1]), take a measurable partition \mathcal P^N of [0,1]=\bigcup_{i=1}^NI_i, and obtain a new function ... • 3,298 4 votes 1 answer 133 views Proof request: Extending vector valued function of unit outer normals to the whole space \Bbb R^n. I was wondering if anyone could provide a proof or a reference of the following proposition: Let C be an open subset of \mathbb{R}^n with C^2 boundary so that the unit outer normal vector v(x) ... • 500 3 votes 1 answer 196 views \sum_{n=1}^\infty 2^{-n} f_n has unbounded variation on every [a,b] \subset [0,1] with a < b Consider n\in \Bbb N, and define a piecewise linear function f_n: [0,1] \to \Bbb R so that the graph of f_n contains the points \{(k \cdot 2^{-n}, (-1)^k)\}_{0\le k\le 2^n}. Define f: =\sum_{... 2 votes 1 answer 60 views Why if p-th variation is finite, then q-th variation is also finite for q>p? Consider function L^p(X) and L^q(X) space with p<q where X is a finite measure space. Then from Holder, I have L^q(X) embedding into L^p(X). In particular, if \|f\|_q<\infty, then ... • 8,520 0 votes 0 answers 145 views What is the positive variation of a function? I am having trouble understanding what the positive variation of a function \frac{ 1}{2} ( T_F + F) represents. (Where, in this case, T_F is the total variation of F from [0,x].) For consider ... • 302 0 votes 0 answers 71 views How to show f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases} is of bounded variation? Prove or disprove: f(x) = \begin{cases} \frac{\sin(x)}{x}, & \text{when } x \neq 0,\\ 0, & \text{when } x = 0. \end{cases} is of bounded variation on [0,1] and \sup_{p \in P} V(f,p)... • 1,125 0 votes 0 answers 91 views Let f be a real-valued, everywhere differentiable function on [0,1]. Suppose f \in BV[0,1]. Prove f \in AC[0,1]. [duplicate] I tried to prove this by showing that the derivative f' must be bounded. But I am not sure this is true, as the Extreme Value Theorem cannot be invoked (f' is not guaranteed to be continuous) nor ... • 302 4 votes 0 answers 91 views Riemann-Stieljes integration and Total variation Let \alpha:[a,b]\to\mathbb{R} be a function of bounded variation on [a,b] and f:[a,b]\to\mathbb{R} a bounded function. It is well known that if f is Riemann-Stieljes integrable respect to \... • 6,620 2 votes 1 answer 91 views L^1 equicontinuity for functions with uniformly bounded total variation I've been trying to prove that the space BV[0, 1] of elements of L^1 [0, 1] with bounded variation, is compactly embedded in L^1[0, 1], using the Fréchet-Kolmogorov theorem. For context, a ... • 186 1 vote 1 answer 64 views Total variation of a class in L^1 [0, 1] and lower semicontinuity Consider a function f \in \mathcal L^1 [0, 1]: we define the total variation of f as usual by$$ V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |, $$where the supremum is taken ... • 186 1 vote 0 answers 35 views boundedness of signed measures Let us consider signed charges, these are finitely additive signed measures (I suppose this would also work with sigma additive signed measures). We work on a measure space (\Omega, \mathcal{A}), ... • 351 1 vote 0 answers 51 views How to prove by contradiction that a function of bounded variation is bounded I would like to prove that a function defined [a,b] of bounded variation is bounded by contradiction : suppose that f is of bounded variation while note bounded and come up with a contradiction. ... • 1,339 1 vote 0 answers 31 views Decay rate of a functional norm. I’ve been working on a project lately that has resulted in needing a following fact: if \chi_{B_r(x)} denotes the characteristic function of the ball of radius r about x in \mathbb{R}^2, then ... • 468 0 votes 1 answer 30 views A norm-decreasing and right-continuous mapping f: [0,1] \to H is of bounded-variation with respect to the norm of H? Let H be a separable Hilbert space and f:[0,1] \to H be a norm-decreasing map in the sense that $$\lVert f(t_2) \rVert_H \leq \lVert f(t_1) \rVert_H$$ if 0 \leq t_1 \... • 7,715 1 vote 0 answers 25 views Proving that for u \in BV([a,b]), \frac{d}{dx} (\text{Var}_{[a,x]} u) = |u'(x)|. I am trying to work through Exercise 2.29 of Leoni's A First Course in Sobolev Spaces, which gives an alternative proof of the theorem in the title in the real-valued case. The exercise is broken up ... • 1,477 2 votes 1 answer 139 views Conditions on Radon measure for it to be a measure of a BV function Functions of bounded variation (BV) on \mathbb{R}^d are functions with$$ \|u\|_{BV} := \|u\|_{L^1(\mathbb{R}^d)} + TV(u) $$finite, where$$ TV(u) := \sup\left\{ \int_{\mathbb{R}^d}u(x)div(\phi)(... 6 votes 1 answer 371 views Is$x^a $sin$(x^{-a})$Hölder continuous? I'm currently trying the following Exercise #11 of Chapter 3 in Stein's Real Analysis. Exercise 11. If$a, b>0$, let $$f(x)=\begin{cases} x^a \sin \left(x^{-b}\right) & \text { for } 0<x \... • 769 1 vote 0 answers 32 views Bounded variation of derivatives when fitting a function from one interval into another I pose this question to seek to prove a statement like the following: Let f(x) and g(x) be continuous on the interval [0, 1], and let h(y)=f((y+1)/2) where -1\le y\le 1. If f^{(r)} is of ... • 941 5 votes 1 answer 112 views Riemann-Stieltjes integral with respect to functions equal almost everywhere [closed] Let f, g be Lebesgue integrable, real-valued functions on [0, 1] with bounded variation, and let \phi : [0, 1] \to \mathbb R be continuous. Assuming that f = g almost everywhere, does one have ... • 186 0 votes 1 answer 52 views The sequence of function f_k(x) = \frac{x^k}{(1+x)^2},~x\ge 0 For non-negative integers k\geq 1 define$$ f_k(x) = \frac{x^k}{(1+x)^2},~x\geq 0. $$Which of the following statements are true? For each k, f_k is a function of bounded variation on ... • 617 0 votes 0 answers 66 views Dealing with preimages of a function of bounded variation The problem is the following: If f is a real-valued function of bounded variation on an interval [a,b] \subset \mathbb{R} . Show that the set$$ A=\{y \in \mathbb{R} : \{x \in [a, b] : f(x)=y \}\:... • 58 2 votes 0 answers 46 views Variation of a Brownian Motion Let$B_t$be a standard brownian motion. The variation is not bounded on any interval. What if I can only detect variation of a,ie I only observe the process$[B/a]_t$. Then this is a bounded ... • 411 0 votes 1 answer 111 views Show that$\forall\varepsilon>0$, there exists a bounded continuous function$g(x)$on$E$, such that$m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$. Let$E$be a measurable subset of$\mathbb{R}^n$, and$mE<\infty$.$f(x)$is a bounded measurable function almost everywhere. I want to find such a function$g$. For$n=1$, i.e.,$E\in\mathbb{R}$.$...
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A metric measure space is a tuple $(X,d,\mu)$ where $(X,d)$ is a metric space and $\mu$ is a measure on $(X,\mathcal{B}_X)$ ($\mathcal{B}_X$ is the notation for the Borel $\sigma$-algebra associated ...
As we know, the Weierstrass function defined by $$f(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x), 0<a<1,b \in 2\mathbb N+1,ab > 1+\frac{3\pi}{2}$$ is differentiable nowhere (see here for a ...