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By Bogdan Dumitrescu

This ebook gathers the most fresh effects on optimistic trigonometric polynomials inside of a unitary framework. The e-book has elements: thought and functions. the idea of sum-of-squares trigonometric polynomials is gifted unitarily in line with the idea that of Gram matrix (extended to Gram pair or Gram set). The purposes half is prepared as a set of comparable difficulties that use systematically the theoretical effects.

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21. 71). Proof. 73) holds, then it is clear that R(ω) ≥ ∆ 0 for any ω. 73). e. n ˜ = 1), with complex coefficients. 73), we have    1 q00 q10 q20 R(ω) = [1 cos ω sin ω]  q10 q11 q21   cos ω  . q20 q21 q22 sin ω By using simple trigonometric expressions like (cos ω)2 = (1 + cos 2ω)/2, we obtain q11 q22 + ) + 2q10 cos ω + 2q20 sin ω R(ω) = (q00 + 2 2 q11 q22 − ) cos 2ω + q21 sin 2ω. 74) 2 2 In the particular case where the polynomial is R(z) = (2 + j)z −2 + (3 − j)z −1 + 9 + (3 + j)z + (2 − j)z 2 , 44 POSITIVE TRIGONOMETRIC POLYNOMIALS on the unit circle we have R(ω) = 9 + 6 cos ω − 2 sin ω + 4 cos 2ω + 2 sin 2ω.

89) ˜ ω]T . χs (ω) = [sin ω . . 90) and With these bases, a Gram parameterization of R(ω) is possible, using two Gram matrices. 28 Let R ∈ Rn [z] be a trigonometric polynomial of order n = 2˜ n. The polynomial is nonnegative if and only if there exist positive semidefinite matrices Q ∈ R(˜n+1)×(˜n+1) and S ∈ Rn˜ טn such that R(ω) = χTc (ω)Qχc (ω) + χTs (ω)Sχs (ω). 91) 48 POSITIVE TRIGONOMETRIC POLYNOMIALS We name (Q, S) a Gram pair associated with R(ω). Proof. 91) holds, it results that ∆ 0 and R(ω) ≥ 0.

If P (t) ≥ 0 for t ≥ 0, then P˘ (t) ≥ 0 for t ∈ [−1, 1]. 54) P˘ (t) = (1 − t)A(t)2 + (1 + t)B(t)2 . 56) where m = n − n ˘ . 54) √ holds, then F (t) =√A(t) and G(t) = 2B(t). 53). 53). 15 Consider the bilinear transform z= 1 + jt (1 + jt)2 = . 58) are equivalent to t = tan(ω/2). 7) ˜ ( z+z −1 ) + R(z) = U (z) + jV (z) = U 2 z−z −1 2j −1 · V˜ ( z+z2 ), 18 POSITIVE TRIGONOMETRIC POLYNOMIALS ˜ , V˜ ∈ R[z] and deg U ˜ = n, deg V˜ = n − 1 (one of the degrees could be with U smaller, but this is irrelevant).

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