S u p x i n [ a , b ] | P n ( x ) − f ( x ) | < ϵ

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In summary, the conversation discusses the possibility of showing that f(x)=0 in the interval [a,b] without using the Weierstrass approximation theorem. The speaker suggests looking up the theorem online and finding a proof that uses it, but also mentions that it is not an easy task. They also mention the need for f(x) to be bounded on the interval in order for the proof to work.
  • #1
feerrr
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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
 
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  • #2
Are you able to express that in a way that let's other people understand what you are asking?
 
  • #3
f in C[a,b] and n=0,1,2,...
we have : x^n*f(x)dx=0 (integral from a to b )
im trying to show that f(x)=0 in [a,b]
can i use : sup x in [a,b] |Pn(x) - f(x) | < ϵ ( Pn(x)=a0+a1x+...+anx^n ) ? and why ?
 
  • #4
I don't follow that you can go straight to that.

Do you know the Weierstrass approximation theorem?

I suspect this is hard to prove from first principles.
 
  • #5
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
 
  • #6
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
What I suggest is to look up the theorem on line and find a proof using it. It's still not easy.

I can't immediately see another way.

Also, you need to know or prove that ##f## is bounded on ##[a,b]##.
 

FAQ: S u p x i n [ a , b ] | P n ( x ) − f ( x ) | < ϵ

What is the purpose of the equation "S u p x i n [ a , b ] | P n ( x ) − f ( x ) | < ϵ"?

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].

What does "P n ( x )" represent in the equation?

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.

How is the value of ϵ determined in this equation?

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.

Can this equation be used for any type of function?

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.

How is this equation useful in scientific research?

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.

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