l-1 summability of multiple Fourier integrals and positivity

Abstract

Let f∈L1(ℝd), and let fˆ be its Fourier integral. We study summability of the l-1 partial integral S(1)R, d(f; x)= ∫[mid ]v[mid ][les ]Reiv·xfˆ(v)dv, x∈ℝd; note that the integral ranges over the l1-ball in ℝd centred at the origin with radius R>0. As a central result we prove that for δ[ges ]2d−1 the l-1 Riesz (R, δ) means of the inverse Fourier integral are positive, the lower bound being best possible. Moreover, we will give an l-1 analogue of Schoenberg's modification of Bochner's theorem on positive definite functions on ℝd as well as an extention of Polya's sufficiency condition.

Criteria of Pólya type for radial positive definite functions

“…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).…”

“…Thus, Koldobskiǐ's [12] and Zastavnyi's [34] solution to the Schoenberg problem implies that k n,α (λ) is finite if (i) n = 1, α ∈ (0, ∞], λ ∈ (0, 2); (ii) n ≥ 2, α ∈ (0, 2), λ ∈ (0, α]; (iii) n ≥ 2, α = 2, λ ∈ (0, 2); and (iv) n = 2, α ∈ (2, ∞], λ ∈ (0, 1]; and infinite otherwise. Numerical values of k n,α (λ) are known in special cases only: k n,2 (1) = (n + 1)/2 (Golubov [10]); k 2,2 (1/2) = 1 (Pasenchenko [23]); k n,1 (1) = 2n − 1 (Berens and Xu [3]); and k 2,∞ (1) = 3. Furthermore, various estimates are known.…”

“…This result again reduces to Pólya's theorem if n = 1. It was established independently by Berens and Xu [3] and Gneiting [5]. Finally, the criteria of Zastavnyi [35,36] and Gneiting [8] stem from the facts that k 2,α (1) ≤ 3 if α ∈ [1, ∞], and k 2,2 (1/2) = 1, respectively.…”

Summability of Product Jacobi Expansions

“…If : j =; j =&1Â2, then $ 0 =0 and the condition $>$ 0 becomes simply $>0. This extremal case corresponds to the l 1 summability of the multiple Fourier series or integral (see [5,6]), which is in sharp contrast with the usual radial (that is, l 2 ) means of the multiple Fourier series or integrals (cf. [19]).…”

On a Theorem of T. Gneiting on α-Symmetric Multivariate Characteristic Functions

“…The set of all these functions will be denoted by 8 d (:). Functions of type (1) are called radial with respect to the l : -norm, or :-radial for short. Due to Bochner's famous theorem :-symmetric characteristic functions can be interpreted as positive definite :-radial functions.…”