Inequalities for Positive Definite Functions
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ualities for Positive Definite Functions V. P. Zastavnyi1* 1
Donetsk National University, Donetsk, 83114 Ukraine
Received February 24, 2020; in final form, June 16, 2020; accepted July 1, 2020
Abstract—Positive definite kernels and functions are considered. The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel. It is shown that Ingham’s inequality (and, in particular, Hilbert’s inequality) is, essentially, the main inequality for the positive definite function sin(πx)/x on R and for a system of integer points. Using the main inequality, we prove new inequalities of Krein–Gorin type and Ingham’s inequality. DOI: 10.1134/S0001434620110218 Keywords: positive definite kernels and functions, Ingham’s inequality, Hilbert’s inequality, Krein’s inequality, Weil’s inequality, Gorin’s inequality.
1. INTRODUCTION Let G be a set. A function K : G × G → C is called a positive definite kernel on G × G if, for any natural number n ∈ N, for any collections {xk }nk=1 ⊂ G of points and {ck }nk=1 ⊂ C of numbers, the following inequality holds: n
ck cp K(xk , xp ) ≥ 0.
(1.1)
k,p=1
The set of all such kernels will be denoted by the symbol Φ(G × G). A kernel K ∈ Φ(G × G) is said to be strictly positive definite if inequality (1.1) is strict under the condition that all the points in {xk }nk=1 are pairwise distinct (i.e., xk = xp for k = p) and the numbers in {ck }nk=1 are not all zero (i.e., |c1 | + · · · + |cn | > 0). Let G be a group (not necessarily Abelian) with group operation +. A function f : G → C is said to be positive definite on G if the function K(x, y) := f (x − y) is a positive definite kernel on G × G. The set of all such functions will be denoted by the symbol Φ(G). For n = 2, x1 = x, and x2 = y, inequality (1.1) for K ∈ Φ(G × G) is of the form a1,1 |c1 |2 + a1,2 c1 c2 + a2,1 c2 c1 + a2,2 |c2 |2 ≥ 0,
c1 , c2 ∈ C,
(1.2)
where a1,1 = K(x, x), a1,2 = K(x, y), a2,1 = K(y, x), and a2,2 = K(y, y). In (1.2), we take c1 = 1 and c2 = 0. Then a1,1 ≥ 0. Similarly, a2,2 ≥ 0. Therefore, a1,2 c1 c2 + a2,1 c2 c1 ∈ R
for any
c1 , c2 ∈ C.
In particular, for c1 c2 = 1 and c1 c2 = i, we obtain 2a := a1,2 + a2,1 ∈ R and 2b := (a1,2 − a2,1 )i ∈ R, respectively. Therefore, a1,2 = a − bi
and
a2,1 = a + bi = a1,2 .
where φ ∈ R. As proved above, a2,1 = |a1,2 |e−iφ . In (1.2), we take c1 = t ∈ R and Let a1,2 = |a1,2 c2 = eiφ . Then, for any t ∈ R, the following inequality holds: |eiφ ,
a1,1 t2 + 2|a1,2 |t + a2,2 ≥ 0. *
E-mail: [email protected]
791
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ZASTAVNYI
Therefore, |a1,2 |2 ≤ a1,1 a2,2 . Thus, inequality (1.2) immediately implies the following simple and well-known properties of positive definite kernels: if K ∈ Φ(G × G), then K(x, x) ≥ 0,
K(y, x) = K(x, y),
|K(x, y)|2 ≤ K(x, x)K(y, y)
for all x, y ∈ G.
In the theory of positive definite kernels and functions, an important role is played by the following Krein inequality (see, e.g., [1, Chap. IV, Sec. 1]): if K ∈ Φ(G × G), then,
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