Positive and Negative Definite Kernels on Trees

Wojciech Młotkowski
Additional contact information
Wojciech Młotkowski: Instytut Matematyczny Uniwersytetu Wrocławskiego

A chapter in Harmonic Analysis and Discrete Potential Theory, 1992, pp 107-110 from Springer

Abstract: Abstract Let X= (V, E) be a tree with the set V of vertices and the set E of edges. For any x ∈ V we will denote by N(x) the neighbourhood of x, i.e. the set v ∈ V: d(v, z) ≤ 1. Suppose that for any x ∈ V we have a fixed positive definite matrix A(x) = (a(v, x, w)) v, w∈N(x) such that a(v, x, v) = 1 for any v ∈ N(x). We define the kernel φ: V × V → C in the following way: if x, y ∈ V and [x, y] = x 0 = x, x 1, x 2,...,x n = y ⊂ V is the geodesic from x to y, then we put $$\varphi (x,y) = \prod\limits_{i = 0}^n {a(x_{i - 1} ,x_i ,x_{i + 1} ),} $$ where, by definition, x −1 = x 0 = x and x n +1 = x n = y. In particular φ(x, x) = 1. $$\beta (x,y) = \prod\limits_{i = 0}^{n - 1} {a(x_{i - 1} ,x_i ,x_{i + 1} ),} $$ α(x,x) = 1, which will help us in computations. Note that for any i ∈ {0,1,2,..., n} we have 1 $$\varphi (x,y) = \beta (x,x_i )a(x_{i - 1} ,x_i ,x_{i + 1} )$$

Keywords: Harmonic Analysis; Scalar Product; Complex Function; Potential Theory; Pairwise Disjoint (search for similar items in EconPapers)
Date: 1992
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-2323-3_10

Ordering information: This item can be ordered from
http://www.springer.com/9781489923233

DOI: 10.1007/978-1-4899-2323-3_10

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().