# bounded variation

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##### 1: 1.4 Calculus of One Variable
###### Functions of BoundedVariation
1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{j})-f(x_{j-1})|,$
If $\mathcal{V}_{a,b}\left(f\right)<\infty$, then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)=\mathcal{V}_{a,x}\left(f\right)$ and $h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$. …
##### 2: 1.8 Fourier Series
Suppose that $f(x)$ is continuous and of bounded variation on $[0,\infty)$. …
##### 3: 1.14 Integral Transforms
Suppose that $f(t)$ is absolutely integrable on $(-\infty,\infty)$ and of bounded variation in a neighborhood of $t=u$1.4(v)). … If $f(t)$ is absolutely integrable on $[0,\infty)$ and of bounded variation1.4(v)) in a neighborhood of $t=u$, then … If $\int^{\infty}_{0}|f(t)|\,\mathrm{d}t<\infty$, $g(t)$ is of bounded variation on $(0,\infty)$ and $g(t)\to 0$ as $t\to\infty$, then … Suppose the integral (1.14.32) is absolutely convergent on the line $\Re s=\sigma$ and $f(x)$ is of bounded variation in a neighborhood of $x=u$. …
##### 4: 10.23 Sums
provided that $f(t)$ is of bounded variation1.4(v)) on an interval $[a,b]$ with $0. …
##### 5: 10.43 Integrals
• (b)

$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\infty)$, and each of the following integrals

• ##### 6: 18.39 Physical Applications
For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha=\beta=0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …
##### 7: 10.40 Asymptotic Expansions for Large Argument
Bounds for $\mathcal{V}_{z,\infty}\left(t^{-\ell}\right)$ are given by …
##### 8: 10.17 Asymptotic Expansions for Large Argument
Bounds for $\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)$ are given by … The bounds (10.17.15) also apply to $\mathcal{V}_{z,-i\infty}\left(t^{-\ell}\right)$ in the conjugate sectors. …
##### 9: 2.3 Integrals of a Real Variable
In both cases the $n$th error term is bounded in absolute value by $x^{-n}\mathcal{V}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator $\mathcal{V}_{a,b}$ is defined by …
##### 10: 2.8 Differential Equations with a Parameter
In addition, $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{1}\right)$ and $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{n}\right)$ must be bounded on $\mathbf{\Delta}_{j}(\alpha_{j})$. …