Prove that the family of all polynomials of degree < N with co- efficients in (-1, 1] is uniformly bounded and uniformly equicon- tinuous on any compact interval.
Prove that the family of all polynomials of degree < N with co- efficients in (-1, 1] is uniformly bounded and uniformly equicon- tinuous on any compact interval.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Prove that the family of all polynomials of degree ≤ \( N \) with coefficients in \([-1, 1]\) is uniformly bounded and uniformly equicontinuous on any compact interval.
**Explanation:**
This statement is a mathematical problem related to the properties of polynomials. It discusses a family of polynomials where:
- The degree of each polynomial is at most \( N \).
- The coefficients of the polynomial are bounded within the interval \([-1, 1]\).
The goal is to demonstrate that this family of polynomials is:
1. **Uniformly Bounded**: There exists a constant \( M \) such that for every polynomial \( P \) in the family, the absolute value of \( P(x) \) is less than or equal to \( M \) for all \( x \) within the compact interval.
2. **Uniformly Equicontinuous**: For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for every polynomial \( P \) in the family and for all \( x, y \) within the compact interval, if the absolute difference \(|x - y| < \delta\), then \(|P(x) - P(y)| < \epsilon\).
These properties are important in the context of analysis and ensure that the polynomials behave nicely across any compact interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad32dd67-a35a-464e-85b6-24f00339bb54%2Fa8832499-6fc6-447f-906f-13bc942d21e6%2F3fdu5cp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Prove that the family of all polynomials of degree ≤ \( N \) with coefficients in \([-1, 1]\) is uniformly bounded and uniformly equicontinuous on any compact interval.
**Explanation:**
This statement is a mathematical problem related to the properties of polynomials. It discusses a family of polynomials where:
- The degree of each polynomial is at most \( N \).
- The coefficients of the polynomial are bounded within the interval \([-1, 1]\).
The goal is to demonstrate that this family of polynomials is:
1. **Uniformly Bounded**: There exists a constant \( M \) such that for every polynomial \( P \) in the family, the absolute value of \( P(x) \) is less than or equal to \( M \) for all \( x \) within the compact interval.
2. **Uniformly Equicontinuous**: For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for every polynomial \( P \) in the family and for all \( x, y \) within the compact interval, if the absolute difference \(|x - y| < \delta\), then \(|P(x) - P(y)| < \epsilon\).
These properties are important in the context of analysis and ensure that the polynomials behave nicely across any compact interval.
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