Functions of Bounded Variation

In summary, under the given conditions, the function f is of bounded variation if and only if there is a supremum of the constants so that they can't go on vertically to infinitum.
  • #1
jdcasey9
28
0

Homework Statement


Given a sequence of scalars (cn) and a sequence of distinct points (xn) in (a, b), define f(x) = cn if x = xn for some n, and f(x) = 0 otherwise. Under what condition(s) is f of bounded variation on [a,b]?


Homework Equations


Vbaf = supp([tex]\Sigma[/tex]lf(ti) - f(ti-1)l< +inf, then f is of Bounded Variation.


The Attempt at a Solution



My understanding of the question is that we have points at x in a vertical line and points along y=0 at every other value of x. So the only way that this function is of bounded variation is if there is a supremum of the constants so that they can't go on vertically to infinitum. If we merely say that they are bounded above we won't have bounded variation, so we must say that the supremum is a member of the set of cn's.
 
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  • #2
No, you're on the wrong track. The total variation [tex]V_a^b f = \sup_P \sum_j |f(x_j+1) - f(x_j)|[/tex] is the supremum of all possible sums of jumps in the value of [tex]f[/tex] where the function is measured at finitely many points. So you should answer two questions:

1. What can a jump of the value of [tex]f[/tex] from one point to another look like?
2. What analytic idea should you use to think about the "supremum of all finite sums of something"?

You can't answer this question by examining the values [tex]c_n[/tex] one by one; there is a sense in which you must consider them collectively.
 
  • #3
So, we need to use norms? If we add the assumption of completeness, we will satisfy bounded variation.

Since we have Thm: BV[a,b] is complete under llfllBV = lf(a)l + vbaf
and
Lemma: llfllinf [tex]\leq[/tex] llfllBV
then we can prove that llfllBV is complete and therefore the supremum of f is convergent?
 
  • #4
jdcasey9 said:
So, we need to use norms? If we add the assumption of completeness, we will satisfy bounded variation.

Since we have Thm: BV[a,b] is complete under llfllBV = lf(a)l + vbaf
and
Lemma: llfllinf [tex]\leq[/tex] llfllBV
then we can prove that llfllBV is complete and therefore the supremum of f is convergent?

The sentences "If we add the assumption of completeness, we will satisfy bounded variation" and "we can prove that [tex]\|f\|_{BV}[/tex] is complete and therefore the supremum of [tex]f[/tex] is convergent" don't make any sense to me.

You do not need to think about the BV-norm or completeness to solve this problem -- you can apply the definition of bounded variation rather directly.
 
  • #5
Ok, to answer your questions:
1. A movement up or down along the x-value.
2. I'm not sure, we can get the supremum of all finite sums of something by taking the l-inf norm of the sum of all of them.
 

What is meant by "Functions of Bounded Variation"?

"Functions of Bounded Variation" refer to a mathematical concept used to measure the smoothness of a function. It is a measure of how much a function varies over a given interval and is often used in the study of calculus and analysis.

How is the bounded variation of a function calculated?

The bounded variation of a function is calculated by taking the sum of the absolute values of the differences between consecutive values of the function over a given interval. This measures the total amount of change in the function over that interval.

What is the significance of bounded variation in mathematics?

Bounded variation is an important concept in mathematics as it helps to classify functions and understand their properties. It is closely related to the concept of continuity and is used to prove the existence of solutions to certain equations.

Can a function have unbounded variation?

Yes, a function can have unbounded variation if the differences between consecutive values of the function become arbitrarily large. This means that the function does not have a finite measure of smoothness and may have very abrupt changes or discontinuities.

What are some real-world applications of bounded variation?

Bounded variation has several real-world applications, such as in signal processing, where it is used to analyze and filter signals. It is also used in finance to model the fluctuations of stock prices. In physics, it is used to analyze the motion of particles and in economics, it is used to model the behavior of markets.

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