1
Introduction
▶
1.1
Significance and context
1.2
Computational illustration: N = 5
1.3
Proof strategy
2
Preliminaries
▶
2.1
Chebyshev polynomials
2.2
Roots of unity and discrete Fourier analysis
2.3
Power sums and elementary symmetric polynomials
3
Discrete Orthogonality Relations
▶
3.1
Sums of cosines at rotated roots
3.2
Chebyshev angle sums
4
Power Sum Invariance
▶
4.1
Strategy via binomial expansion
4.2
Computational examples
4.3
General power sum invariance
5
From Power Sums to Polynomial Coefficients
▶
5.1
Newton’s identities: the algebraic bridge
5.2
Why the constant term varies
6
Proof of Main Theorem
▶
6.1
Chebyshev roots and their power sums
6.2
Power sum equality
6.3
Completion of the proof
6.4
Explicit verification for small N
Dependency graph
Chebyshev Circles Blueprint
Eric Vergo
1
Introduction
1.1
Significance and context
1.2
Computational illustration: N = 5
1.3
Proof strategy
2
Preliminaries
2.1
Chebyshev polynomials
2.2
Roots of unity and discrete Fourier analysis
2.3
Power sums and elementary symmetric polynomials
3
Discrete Orthogonality Relations
3.1
Sums of cosines at rotated roots
3.2
Chebyshev angle sums
4
Power Sum Invariance
4.1
Strategy via binomial expansion
4.2
Computational examples
4.3
General power sum invariance
5
From Power Sums to Polynomial Coefficients
5.1
Newton’s identities: the algebraic bridge
5.2
Why the constant term varies
6
Proof of Main Theorem
6.1
Chebyshev roots and their power sums
6.2
Power sum equality
6.3
Completion of the proof
6.4
Explicit verification for small N