# Tagged Questions

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### Showing uniqueness

I have a problem which asks to show that a function $f$ of bounded variation can be expressed uniquely except for addition of constants as the sum of an absolutely continuous function and a singular ...
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### Perturbations of a function of bounded variation

Suppose $f,g:[0,1]\rightarrow[0,1]$, $g>0$, and $f/g$ is of bounded variation. If $f_n, f \in C^1[0,1]$ and $f_n \rightarrow f$ in $C^1$. Does it follow that $\exists N$ such that $f_n/g$ is of ...
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### Functions of bounded variations and Riemann-Stieltjes integral

In my work, I used recently the classical Riesz theorem. It has lead me to study functions of bounded variations and Riemann-Stieltjes integrals. Unfortunately, even if there exist a lot of books and ...
Curve $C$ in the plain, \begin{align} C:= \begin{cases} x=\phi(t) \\ y=\psi(t) \end{cases} \quad, \quad a \le t \le b \end{align} The graph of C is $\{(x, y): x=\phi(t), y=\psi(t), a\le t\le b\}$ $\... 0answers 39 views ### When is the Stieltjes integral of bounded variations? I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let$f$be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ... 0answers 46 views ### Wrong result: a continuous function has zero$p$-variation, for every$p$. Where's the error? Let$\Pi_n$be a sequence of partitions with$|\Pi_n| \to 0$. Then the$p$-variation of a continuous function$g$along the partitions$\Pi_n$is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, \... 0answers 44 views ### Relationship between functions of bounded variation and signed measures Let f be a left continuous function of bounded variation on [a,b] s.t. f(a)=0. f can be written as the difference f=f_1-f_2 of two non-decreasing functions on [a,b]. f_1=\frac{1}{2}(v(x)... 0answers 25 views ### Guess quadratic variation of 2 processes, with same/different BM. I have 2 processes with stochastic parts R, S.$$dR = \mu_1 dt + \sigma_1 dW_{t1}dS = \mu_2 dt + \sigma_2 dW_{t2}$$I am trying to show what precisely quadratic variation [R,S] means for 2 ... 0answers 16 views ### RCLL (cadlag) But Not Bounded Variation In probability theory, we often require that functions and/or processes be RCLL (i.e. cadlag) and of bounded variation (usually bounded on any finite interval). I'm having trouble coming up with an ... 0answers 32 views ### Bounding fluctuations on a random variable I have some discrete random variable w that has values (in decreasing order) \underline{w}^\downarrow = \left( w_1, w_2, \dots , w_d \right) with corresponding probabilities \underline{x}=(x_1, ... 0answers 124 views ### Absolutely continuous iff continuous of bounded variation I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ... 0answers 473 views ### Absolute continuity implies bounded variation Let f be absolutely continuous on [a,b]. I want to prove that f is of bounded variation. I am reading Royden and Fitzpatrick so they use the following notations: Let P=\{x_0=a,x_1,\ldots,x_{n-1}... 0answers 73 views ### p-variation of semimartingales Does every (particularly continuous) semi-martingale have bounded 2+\epsilon-variation for all \epsilon>0? Note that I am not asking, whether they have finite quadratic variation - that is ... 0answers 70 views ### Counterexample for Trace operator in BV space. Suppose \Omega\subset R^N is open bounded with Lipschitz boundary. Then for u\in W^{1,p}(\Omega), there exists a linear operator T: W^{1,p}(\Omega)\to L^{p}(\partial\Omega) such that the ... 0answers 47 views ### Distributional Representation of Perimeter in Chan-Vese While re-implementing the classic Chan-Vese Algorithm for image segmentation, I stumbled upon the following statement, which I have problems to understand: Let H: \mathbb{R} \to \mathbb{R} be the ... 0answers 64 views ### Change of variables formula with integrator of bounded variation Let G be be continuous with bounded variation on finite intervals. If f is continuous then it is well known that \int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx. How general can f be so that ... 0answers 70 views ### Question on Bounded Variation involving partitions If f\colon[a,b]\rightarrow \mathbb{R} and P=(a=x_0, x_1, ...,x_n=b) is any partition of [a,b] let v_f(P) be defined as$$v_f(P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| $$and the total variation ... 0answers 315 views ### Every polynomial bounded on [a,b] is of bounded variation. Question: Prove that every polynomial bounded on the compact interval [a,b] is of bounded variationMy proof: Let f(x) be a bounded polynomial of n degree. Then f '(x) is also a polynomial bounded ... 0answers 97 views ### How to show that the first derivative is bounded in a function $$y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber$$ How to show in above function the ... 0answers 274 views ### Solving \int_{0}^{2\pi} \cos (x) \cos (z(x)) \, dx with a bounded version of z(x)=a \cos (x)+b I am trying to solve the following equation:$$\int_{0}^{2 \pi } \cos (x) \cos z(x) \, dx \hspace{20 mm} (1)$$where z(x) is a bounded version of z(x)=a \cos (x)+b that reads as:$$z(x)=\min\... 0answers 102 views ### What is Mountain Pass theorem? The title says it all, what is mountain pass theorem, how does it work is variational problems with doubly differential constraint? 0answers 7 views ### How to indicate which is the best bound for accuracy either the max or min bounds? A Few lists of precipitation data (P-data at different stations) in descending order is used to estimate streamflow at its corresponding stations. For every P data, 2 streamflow values are ... 0answers 13 views ### Proof Function is Bounded/Unbounded How can I prove that the function $$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$ is bounded/unbounded? Note:$\sigma_i\left(t\right)$is ... 0answers 26 views ### Minimum surface area soap film variational principles An axisymmetric soap film$y(x)$is formed between two circular wires at$x = Â±l$. The wires both have radius$r$. Show that the shape that minimises the surface area takes the form $$y(x) = ... 0answers 17 views ### Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland. This is a characterization of the Jordan decomposition of F from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let F\in BV be a real valued function and T_F... 0answers 34 views ### Prove that if f is of bounded variation on [a, b], then f is bounded on [a, b]. Prove that if f is of bounded variation on [a, b], then f is bounded on [a, b]. My professor proved the proposition like the following processes: Choose x between a and b, that is, ... 0answers 47 views ### Can someone prove this bounded variation problem? I am doing my homework, which is solving all even number problems in textbook. Finally, I have almost finished.... but I failed to solve one problem. I am not sure that it is okay if I post the whole ... 0answers 28 views ### Construct a continuous function f with 0\leq f\leq 1 such that \int_a^bf d\alpha\geq\alpha(d)-\alpha(c)-\epsilon Suppose that \alpha is right continuous and increasing. Given \epsilon > 0 and [c,d]\subset[a,b], construct a continuous function f with 0\leq f\leq 1 such that \int_a^bf d\alpha\geq\... 0answers 36 views ### Given f(x) = x^{1/3}, show that f \in BV[0, 1] I'm learning about functions of bounded variations and need some help with this problem: Given f(x) = x^{1/3}, x \in [0, 1] show that f \in BV[0, 1]. My work and thoughts: We know that ... 0answers 25 views ### Show that V_{a}^{b}(X_\mathbb{Q}) = +\infty on any interval [a, b] I'm self-studying the theory of functions of bounded variations from the book of Neal Carother (Real Analysis) and need some help with this problem: Show that V_{a}^{b}(X_\mathbb{Q}) = +\infty ... 0answers 75 views ### Upper bound on successive difference inequalities I would like to know tight upper bounds of the following equations perhaps some might have telescopic sum which can result in very tight bound.In the following equations assume B\geq a_i \geq 0,\... 0answers 40 views ### Total variation function For a function f : [a,b] \times [c,d] \rightarrow R, we define v_f : [a,b] \times [c,d] \rightarrow R as (x,y) \rightarrow V_{f_{[a,x] \times [c,y]}}. Here V_f is the total variation of f ... 0answers 28 views ### Under Given conditions, is f absolutely continuous on [0,1]? 1)Let f: [0,1] \to R be a continuous function such that \mid f(x)-f(y) \mid \le \mid \sqrt{x}-\sqrt{y} \mid for all x,y \in [0,1]. Then is f absolutely continuous on [0,1]? 2) what about ... 0answers 27 views ### Difference of increasing functions differentiable a.e. I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is ... 0answers 19 views ### The normalization of gradient in weak convergence. Given \Omega\subset \mathbb R^2 is open bounded, smooth boundary, u_n\in BV(\Omega) is bounded in BV norm and in addition we have$$0<\inf |u_n|_{TV}\leq \sup |u_n|_{TV}<+\infty$$where$|\...
If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function \$\varphi \in C_c^1(\...