bounded variation

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1: 1.4 Calculus of One Variable

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Functions of Bounded Variation

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1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n|f(x_j)-f(x_{j-1})|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

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See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►If , then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)={𝒱}_{a,x}\left(f\right)$ and $h(x)={𝒱}_{a,x}\left(f\right)-f(x)$ 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 $[0,\mathrm{\infty })$. …

3: 1.14 Integral Transforms

… ►Suppose that $f(t)$ is absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty })$ and of bounded variation in a neighborhood of $t=u$ (§1.4(v)). … ►If $f(t)$ is absolutely integrable on $[0,\mathrm{\infty })$ and of bounded variation (§1.4(v)) in a neighborhood of $t=u$, then … ►If , $g(t)$ is of bounded variation on $(0,\mathrm{\infty })$ and $g(t)\to 0$ as $t\to \mathrm{\infty }$, then … ►Suppose the integral (1.14.32) is absolutely convergent on the line $\mathrm{\Re }s=\sigma$ and $f(x)$ is of bounded variation in a neighborhood of $x=u$. …

4: 10.23 Sums

… ►provided that $f(t)$ is of bounded variation (§1.4(v)) on an interval $[a,b]$ with . …

5: 10.43 Integrals

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

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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 ${𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by …

8: 10.17 Asymptotic Expansions for Large Argument

… ►Bounds for ${𝒱}_{z,\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by … ►The bounds (10.17.15) also apply to ${𝒱}_{z,-\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ 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 $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by …

10: 2.8 Differential Equations with a Parameter

… ►In addition, ${𝒱}_{{𝒬}_j}\left(A_1\right)$ and ${𝒱}_{{𝒬}_j}\left(A_n\right)$ must be bounded on ${𝚫}_j({\alpha }_j)$. …