FUNCTIONS OF BOUNDED VARIATION-II

This lecture focuses on Functions of Bounded Variation (BV).
For a function $f$ and a partition $P$, the total variation $T^b_a(f)$, positive variation $P^b_a(f)$, and negative variation $N^b_a(f)$ are defined as the supremum of the sums $t(P,f)$, $p(P,f)$, and $n(P,f)$, respectively, over all partitions $P$.
$t(P,f) = \sum_{i=1}^{n}|f(x_i)-f(x_{i-1})|$.
$p(P,f) = \sum_{i=1}^{n}(f(x_i)-f(x_{i-1}))^{+}$.
$n(P,f) = \sum_{i=1}^{n}(f(x_i)-f(x_{i-1}))^{-}$.
The relationships between these variations are:
$T^b_a(f) = P^b_a(f) + N^b_a(f)$.
$f(b) - f(a) = P^b_a(f) - N^b_a(f)$.
A function $f:[a,b] \rightarrow \mathbb{R}$ is BV if and only if $f$ is the difference of two monotonic functions.
Monotonic functions are BV, and the sum and difference of BV functions are also BV.
The converse is shown by defining $g(x) = P^x_a(f)$ and $h(x) = N^x_a(f)$, which are monotonically increasing. Since $f(x) = g(x) - (h(x) - f(a))$, $f$ is the difference of two monotonic functions.
If $f:[a,b] \rightarrow \mathbb{R}$ is BV, then $f$ is differentiable almost everywhere. This is because $f$ is the difference of two monotonic functions, $g$ and $h$, which are differentiable almost everywhere.
If $f:[a,b] \rightarrow \mathbb{R}$ is BV, then $f'$ is integrable on $[a,b]$ and $\int_{[a,b]}|f'|dm_1 \le T^b_a(f)$.
If $f \in C^1[a,b]$, then $\int_{[a,b]}|f'|dm_1 \le T^b_a(f)$. Combining this with $T^b_a(f) \le \int_{[a,b]}|f'|dm_1$, it is shown that for $f \in C^1[a,b]$, $T^b_a(f) = \int_{[a,b]}|f'|dm_1$.
The proof for the inequality $\int_{[a,b]}|f'|dm_1 \le T^b_a(f)$ relies on the fact that $x \mapsto P^x_a(f)$ and $x \mapsto N^x_a(f)$ are monotonic and differentiable almost everywhere, leading to $|f'(x)| \le (T^x_a(f))'$ a.e.
The lecture concludes by mentioning the extension of these concepts to vector-valued functions, $f:[a,b] \rightarrow \mathbb{R}^n$.

Видео FUNCTIONS OF BOUNDED VARIATION-II канала Dr P K Chaurasia