A survey of my joint research with Gavin Brown 1

I have been collaborating with Gavin Brown since 1990. Our research mainly concerns the positivity of trigonometric sums and Jacobi polynomial sums. It also concerns multiple trigonometric sums and the convergence of the linear means of multiple Fourier series and Fourier-Laplace series. The present report is a survey on our joint results. §1 Positivity of trigonometric sums In 1990, we considered the sine sums T n (θ) := n ∑ k=1 sin kθ k + α , n ∈ N+. We define three constants λ0, μ0, α0 as follows. The constant λ0 is the solution of the equation (1 + λ)π = tan(λπ) 0 < λ < 1 2 . It is easy to see that λ0 = 0.4302967 · · · is the point at which the function sin λπ 1+λ attains its maximum on the interval (0, 1 2 ). The constant μ0 = 0.8128252 · · · is defined to be the solution of the equation sin μπ μπ = sin λ0π (1 + λ0)π , and α0 = 2.1 · · · is the solution of the equation ∞ ∑ k=1 2k (2k − 1 + α)(2k + α)(2k + 1 + α) = sin λ0π 2(1 + λ0)π . Our results are the following four theorems which were published in [1]. Supported by the NSF of China, #10471010 1 Theorem 1 If −1 < α 6 α0 then T 2n−1(θ) > 0, 0 < θ < π, n ∈ N+. Theorem 2 If −1 < α 6 α0 then T 2n(θ) > 0, 0 < θ < π − μ0π 2n + 0.5 , n ∈ N+. Theorem 3 If α > α0 then there exists an infinite subset N ⊂ N+ such that T 2n−1 ( π − (1 + λ0)π 2n− 1 ) < 0, n ∈ N. Theorem 4 If 0 < γ < μ0 then there exists an α near to but strictly smaller than α0 such that T 2n ( π − γπ 2n + 0.5 ) < 0, for an infinite number of n. Theorem 3 shows that α0 is best possible in Theorem 1. Theorem 4 shows that μ0 is best possible in Theorem 2. The particular case α = 1 has been considered by Brown and Wilson [2]. They obtained the following conclusion: T 1 2n−1(θ) > 0, 0 < θ < π; T 1 2n(θ) > 0, 0 < θ < π − π 2n . These extended the Fejér-Jackson-Gronwall inequality [3], n ∑ k=1 sin kθ k > 0, 0 < θ < π, Later, in 1991 we considered basic cosine sums with Dr. D.Wilson together. The sums we considered are S n(θ) := 1 + n ∑ k=1 k−β cos kθ, n ∈ N+, β > 0, θ ∈ R. Define β0 to be the unique root in (0, 1) of the equation ∫ 3 2 π 0 cos u uβ du = 0. It is easy to check that 0.308443 < β0 < 0.308444. Our main result is 2 Theorem 5[4] For all θ ∈ R and n > 1 we have S0 n (θ) > 0.0376908. It is know (see [5], V.2, 29) that for β < β0 the sums S β n(θ) are not uniformly bounded below. An immediate corollary to Theorem 5 is Theorem 6[4] Let β > β0 and suppose {ak}k=0 is a non-increasing sequence of non-negative numbers satisfying kak > (k + 1)ak+1 for all k > 1. Then n ∑ k=0 ak cos kθ > 0 for all θ ∈ R and n > 0. These results extend the Young’s inequality [6] which is a particular case of Theorem 6 when β = 1 and a0 = 1, ak = k −1 for all k > 1. Remark There is no analogue of Theorem 5 for sine sums. Indeed, for any β < 1, lim sup n→∞ min θ∈R n n ∑ k=1 k−β sin kθ 6 − 2 . L. Vietoris [7] proved in 1958 that if a0 > a1 > · · · > an > 0 and a2k 6 (1 − 1 2k )a2k−1, k > 1, then for n > 1 and θ ∈ (0, π), n ∑ k=1 ak sin kθ > 0 and n ∑