- #1
feerrr
- 3
- 0
why sup x in [a,b] |Pn(x) - f(x) | < ϵ , Pn(x)=a0+a1x+...+anx^n
why f(x)-ϵ<Pn(x)<f(x)+ϵ
why f(x)-ϵ<Pn(x)<f(x)+ϵ
What I suggest is to look up the theorem on line and find a proof using it. It's still not easy.feerrr said:no i don't know the Weierstrass approximation theorem .
can i show that f(x)=0 without using the Weierstrass approximation theorem ? i need some help
The equation represents a mathematical concept known as the supremum norm, which is used to measure the distance between two functions. In this case, it is used to determine the accuracy of a polynomial approximation of a function f(x) on the interval [a,b].
P n ( x ) is a polynomial function of degree n that is used to approximate the original function f(x) on the interval [a,b]. It is also known as the interpolating polynomial.
The value of ϵ is typically chosen by the scientist or mathematician conducting the analysis. It represents the desired level of accuracy for the polynomial approximation. A smaller value of ϵ indicates a higher level of accuracy.
Yes, this equation can be used for any continuous function f(x) on the interval [a,b]. However, the accuracy of the polynomial approximation will vary depending on the complexity of the function and the degree of the polynomial used.
This equation is useful for evaluating the accuracy of polynomial approximations in various scientific fields, such as physics, engineering, and statistics. It allows researchers to determine the level of error in their approximations and make improvements to their methods. It is also a fundamental concept in numerical analysis and approximation theory.