Bernstein's theorem on monotone functions
Bernstein's theorem on monotone functions is a rule that helps us understand how certain kinds of functions behave. A function is a special kind of rule that takes one number as input and gives us another number as output.
Now imagine we have a function that takes in numbers and always gives us a bigger answer if we input a bigger number. This is called a monotone increasing function. On the opposite, if we input a bigger number and get a smaller answer, this is called a monotone decreasing function.
Bernstein's theorem tells us that any function that is monotone increasing or decreasing can be approximated really well by a special type of function called a polynomial. Polynomials are functions that you make by adding up different powers of x, like x, x^2, x^3, and so on.
What Bernstein's theorem says is that no matter how complicated or strange a monotone function might be, we can always find a polynomial that comes really close to it. We say the polynomial “approximates” the function really well or becomes “arbitrarily” close to it. Thinking like a five-year-old, it's like we can find a cookie that looks similar to a cake you really like.
This seems like a very complicated idea but it can be very useful in real life. For example, suppose we are trying to figure out how much food we need to buy for a big party. We can use Bernstein's theorem to estimate how many people will come based on previous parties we have had.
So in simpler terms, Bernstein's theorem tells us that we can always find a polynomial that can give us a really good idea of what any monotone function looks like.
Now imagine we have a function that takes in numbers and always gives us a bigger answer if we input a bigger number. This is called a monotone increasing function. On the opposite, if we input a bigger number and get a smaller answer, this is called a monotone decreasing function.
Bernstein's theorem tells us that any function that is monotone increasing or decreasing can be approximated really well by a special type of function called a polynomial. Polynomials are functions that you make by adding up different powers of x, like x, x^2, x^3, and so on.
What Bernstein's theorem says is that no matter how complicated or strange a monotone function might be, we can always find a polynomial that comes really close to it. We say the polynomial “approximates” the function really well or becomes “arbitrarily” close to it. Thinking like a five-year-old, it's like we can find a cookie that looks similar to a cake you really like.
This seems like a very complicated idea but it can be very useful in real life. For example, suppose we are trying to figure out how much food we need to buy for a big party. We can use Bernstein's theorem to estimate how many people will come based on previous parties we have had.
So in simpler terms, Bernstein's theorem tells us that we can always find a polynomial that can give us a really good idea of what any monotone function looks like.