A note on certain functional determinants
type: Item
instance of: scholarly article
title: A note on certain functional determinants
zbMATH Open document ID: 0762.39011
DOI: 10.1007/BF01834202
published in: Aequationes Mathematicae
publication date: 1993-01-06T00:00:00Z
full work available at URL: https://eudml.org/doc/137477
review text: Given an arbitrary set \(X\), a positive integer \(n\) and a real or complex valued function \(F\) defined on \(X^ n\), the author establishes a necessary and sufficient condition for the representation \((*)\) \(F(x_ 1,\dots,x_ n)=Det[f_ j(x_ k)]_{n\times n}\) on \(X^ n\) with a suitable \(n\)-tuple of functions \(f_ 1,\dots,f_ n\) defined on \(X\). He also proves that any functions \(f_ 1,\dots,f_ n\) occurring in the representation \((*)\) have to be linear combinations of the functions \(F(u_ 1,\dots,u_{j-1},\cdot,u_{j+1},\dots,u_ n)\), \(j\in\{1,\dots,n\}\), where \((u_ 1,\dots,u_ n)\in X^ n\) such that \(F(u_ 1,\dots,u_ n)\neq 0\). Next he describes: 1. functions \(H\) (real or complex) defined on the product \(X\times Y\) of two sets that admit a representation \(H(x,y)=\sum^ n_{j=1}f_ j(x)f_ j(y)\) on \(X\times Y\); 2. functions \(H\) (real or complex) defined on \(X\times X\) that admit a representation \(H(x,y)=\sum^ n_{j,k=1}\alpha_{jk}f_ j(x)f_ k(y)\) on \(X\times X\).
reviewed by: Karol Baron
Mathematics Subject Classification ID: 39B05
Mathematics Subject Classification ID: 26B40
zbMATH DE Number: 85815
zbMATH Keywords: functional equations
zbMATH Keywords: functional determinants
zbMATH Keywords: Casorati determinant
author: J. Šimša
MaRDI profile type: Publication
cites work: Q3992445
cites work: Decomposable functions of several variables
cites work: Sums of products of functions of x times functions of y
cites work: Q3665260
cites work: Finite sums of products of functions in single variables
cites work: Q3785070
OpenAlex ID: W2085502543
description: scientific article; zbMATH DE number 85815