Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Theorem: Uniform Boundedness and Equicontinuity of Polynomial Families**
*Statement:* Prove that the family of all polynomials of degree \( \leq N \) with coefficients in \([-1, 1]\) is uniformly bounded and uniformly equicontinuous on any compact interval.
**Explanation:**
This theorem asserts that polynomials with a restricted degree and coefficients within a specified range maintain certain boundedness and continuity properties when considered over compact intervals. Here are key concepts involved:
1. **Degree \( \leq N \):** The polynomials under consideration have degrees that do not exceed \( N \).
2. **Coefficients in \([-1, 1]\):** Each coefficient of these polynomials is constrained to lie within the interval \([-1, 1]\).
3. **Uniformly Bounded:** There exists a single constant \( M \) such that the absolute value of all polynomials in this family is less than or equal to \( M \) across the entire compact interval.
4. **Uniform Equicontinuity:** All polynomials in this family exhibit continuity behavior that is consistent across the family, with respect to a uniform standard, on any compact interval.
This problem is a classic exercise in the analysis of function families, illustrating concepts such as boundedness and continuity which are crucial for understanding the behavior of functions under limitations and constraints in mathematical analysis.](https://content.bartleby.com/qna-images/question/90cdfdd6-d27f-46af-8b0b-68d8124feaf1/23cc49a7-d197-4968-aa65-34afa3cec94d/yqw9icw_processed.png)
Transcribed Image Text:**Theorem: Uniform Boundedness and Equicontinuity of Polynomial Families**
*Statement:* Prove that the family of all polynomials of degree \( \leq N \) with coefficients in \([-1, 1]\) is uniformly bounded and uniformly equicontinuous on any compact interval.
**Explanation:**
This theorem asserts that polynomials with a restricted degree and coefficients within a specified range maintain certain boundedness and continuity properties when considered over compact intervals. Here are key concepts involved:
1. **Degree \( \leq N \):** The polynomials under consideration have degrees that do not exceed \( N \).
2. **Coefficients in \([-1, 1]\):** Each coefficient of these polynomials is constrained to lie within the interval \([-1, 1]\).
3. **Uniformly Bounded:** There exists a single constant \( M \) such that the absolute value of all polynomials in this family is less than or equal to \( M \) across the entire compact interval.
4. **Uniform Equicontinuity:** All polynomials in this family exhibit continuity behavior that is consistent across the family, with respect to a uniform standard, on any compact interval.
This problem is a classic exercise in the analysis of function families, illustrating concepts such as boundedness and continuity which are crucial for understanding the behavior of functions under limitations and constraints in mathematical analysis.
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