On Positive Definiteness of Some Functions

Criteria of Pólya type for radial positive definite functions

“…The second inequality is immediate because k n,α (λ) is nondecreasing in λ, and the upper estimate is given in Corollary 4 of Zastavnyi [35].…”

“…If we fix λ = 1, then k n,α (1) is the Richards-Askey function which the author introduced in [5]. Zastavnyi [35] established an interesting connection to Schoenberg's [28] classical question whether ϕ(t) = exp(−t λ ) is an element of Φ n (α). Specifically, he showed that if λ ∈ (0, 2), then k n,α (λ) is finite if and only if ϕ(t) = exp(−t λ ) belongs to the class Φ n (α).…”

“…Furthermore, various estimates are known. Golubov [10] showed that k n,2 (λ) ≥ (n − 1)/2 + λ, and Zastavnyi's [35] Corollary 3 implies that k 2,α (1) ≤ 3 if and only if α ≥ 1. The following result shows that k n,α (λ) grows at least logarithmically, and at most linearly, with n. The crucial upper estimate is due to Zastavnyi [35].…”

“…The problem can be formulated equivalently in terms of summability and positivity for Fourier series and Bessel integrals, and we refer to Kuttner [14], Golubov [10], Misiewicz and Richards [20], Berens and Xu [3], and Zastavnyi [35,36] for these relations. The univariate case, n = 1, has an interesting history and dates back to the works of Wintner [32] and Kuttner [14] (see [9] for a more detailed discussion).…”

“…Zastavnyi [35,36] investigated for which values of λ > 0 and κ > 0 the truncated power function ϕ λ,κ (t) = (1 − t λ ) κ + is an element of the class Φ n (α). He showed that for all n ≥ 1 and α > 0 there exists a nondecreasing but not necessarily finite function k n,α (λ), λ ∈ (0, 2), such that ϕ λ,κ is an element of Φ n (α) if and only if κ ≥ k n,α (λ).…”