Tagged Questions

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questions regarding definition of bounded variation of several variables

$\int_\Omega u(x)\,\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = -\int_\Omega \langle\boldsymbol{\phi}, Du(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)$ (http://en.wikipedia....
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Does the total variation of a function bound its numerical integration error, much like its first derivative?

When estimating the convergence of a Riemann sum to its integral, or equivalently the error in numerical integration, the commonly used bound is by upper bounding it's first derivative (see, for ...
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Bounded variation, difference of two increasing functions

Prove that if $f$ is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions; and the difference of two bounded monotonic increasing functions is a ...
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The limit of a sequence of uniform bounded variation functions in $L_1$ is almost sure a bounded variation function

Let $\{f_n\}$be a sequence of functions on $[a,b]$ that $\sup V^b_a (f_n) \le C$, if $f_n \rightarrow f$ in $L_1$ ,Prove that $f$ equals to a bounded variation function almost every where. I ...
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Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
I find this pretty hard and it would be awesome if someone could help me. The problem is the following (Problem 6/Chapter 3 from S&S's Real Analysis). Suppose $F$ is a bounded measurable ...