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

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California State University, Bakersfield

Signals and Systems

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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.