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Signals and Systems
97
California State University, Bakersfield
Hani Mehrpouyan,
Department of Electrical and Computer Engineering,
California State University, Bakersfield
Lecture 16 (Chebyshev Polynomials)
April 25th, 2013
The material in these lectures is partly taken from the books: Elementary Numerical Analysis”, 3rd Edition, K. Atkinson, W. Han, “Applied Numerical Methods Using MATLAB”, 1st Edition, W. Y. Yang, W. Cao, T.-S. Chung, J. Morris, and
“Applied Numerical Analysis Using MATLAB”, L V. Fausett.
Signals and Systems
98
California State University, Bakersfield
Chebyshev Polynomials� Chebyshev polynomials are used in many parts of
numerical analysis, and more generally, in applications of mathematics.
� For an integer n≥0, define the function
� This may not appear to be a polynomial, but we will show it is a polynomial of degree n. To simplify the manipulation of (1), we introduce
� Then
Signals and Systems
99
California State University, Bakersfield
Signals and Systems
100
California State University, Bakersfield
Chebyshev Polynomials� The triple recursion relation. Recall the trigonometric
addition formulas,
� Let n≥1, and apply these identities to get
� Add these two equations, and then use (1) and (3) to obtain
Signals and Systems
101
California State University, Bakersfield
Chebyshev Polynomials� This is called the triple recursion relation for the
Chebyshev polynomials. It is often used in evaluating them, rather than using the explicit formula (1).
Signals and Systems
102
California State University, Bakersfield
Signals and Systems
103
California State University, Bakersfield
Chebyshev Polynomials� The minimum size property. Note that
� for all n≥0. Also, note that
� This can be proven using the triple recursion relation and mathematical induction.
� Introduce a modified version of Tn(x),
� From (5) and (6),
Signals and Systems
104
California State University, Bakersfield
Signals and Systems
105
California State University, Bakersfield
Chebyshev Polynomials� A polynomial whose highest degree term has a coefficient
of 1 is called a monic polynomial. Formula (8) says the monic polynomial Tn(x) has size 1/2
n−1 on −1≤x≤1, and this becomes smaller as the degree n increases. In comparison,
� Thus, xn is a monic polynomial whose size does not change with increasing n.
� Theorem: Let n≥1 be an integer, and consider all possible monic polynomials of degree n. Then the degree n monicpolynomial with the smallest maximum on [−1, 1] is the modified Chebyshev polynomial e Tn(x), and its maximum value on [−1, 1] is 1/2n−1.
Signals and Systems
106
California State University, Bakersfield
Signals and Systems
107
California State University, Bakersfield
Signals and Systems
108
California State University, Bakersfield
Chebyshev Polynomials� This result is used in devising applications of Chebyshev
polynomials. We apply it to obtain an improved interpolation scheme in the next lecture.
Signals and Systems
109
California State University, Bakersfield
Hani Mehrpouyan,
Department of Electrical and Computer Engineering,
California State University, Bakersfield
Lecture 17 (A Near-Minimax Approximation Method)
April 25th, 2013
The material in these lectures is partly taken from the books: Elementary Numerical Analysis”, 3rd Edition, K. Atkinson, W. Han, “Applied Numerical Methods Using MATLAB”, 1st Edition, W. Y. Yang, W. Cao, T.-S. Chung, J. Morris, and
“Applied Numerical Analysis Using MATLAB”, L V. Fausett.