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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
why f(x)-ϵ<Pn(x)<f(x)+ϵ
S u p x i n [ a , b ] | P n ( x ) − f ( x ) | < ϵ
- Context: High School
- Thread starter feerrr
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Discussion Overview
The discussion revolves around the mathematical concept of approximating a continuous function \( f \) on the interval \([a, b]\) using polynomial functions \( P_n(x) \). Participants explore the implications of the condition \( \sup_{x \in [a,b]} |P_n(x) - f(x)| < \epsilon \) and its relation to proving that \( f(x) = 0 \) on the interval.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions the meaning of the condition \( \sup_{x \in [a,b]} |P_n(x) - f(x)| < \epsilon \) and its implications for the inequalities \( f(x) - \epsilon < P_n(x) < f(x) + \epsilon \).
- Another participant seeks clarification on how to express their question more clearly for others to understand.
- A participant states that if \( f \) is in \( C[a,b] \) and \( n = 0, 1, 2, \ldots \), they have the integral condition \( \int_a^b x^n f(x) \, dx = 0 \) and wonders if they can use the supremum condition to show \( f(x) = 0 \) on \([a,b]\).
- One participant expresses skepticism about directly using the supremum condition and references the Weierstrass approximation theorem, suggesting that proving the statement may be difficult from first principles.
- Another participant admits to not knowing the Weierstrass approximation theorem and asks if it is possible to show \( f(x) = 0 \) without it, requesting assistance.
- A later reply reiterates the lack of knowledge about the Weierstrass approximation theorem and suggests looking it up for a proof, while also noting the need to establish that \( f \) is bounded on \([a,b]\).
Areas of Agreement / Disagreement
Participants express uncertainty regarding the use of the supremum condition to prove \( f(x) = 0 \). There is no consensus on whether the Weierstrass approximation theorem is necessary or if an alternative approach exists.
Contextual Notes
Participants acknowledge the need for additional assumptions, such as the boundedness of \( f \) on \([a,b]\), which may affect the validity of their arguments.
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