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
icon
Related questions
Question

315 6

 

**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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,