Sums of products of functions of x times functions of y
type: Item
instance of: scholarly article
title: Sums of products of functions of x times functions of y
zbMATH Open document ID: 0695.26007
DOI: 10.1016/0024-3795(89)90031-1
author: Hillel Gauchman
author: Lee A. Rubel
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: The authors investigate functions u defined on a connected open non-empty subset of a region G of \(R^ 2\) which admit the representation \[ (1)\quad u(x,y)=\sum^{n}_{k=1}A_ k(x)B_ k(y), \] where \(A_ k\) and \(B_ k\) are defined on the projections of G onto x-axis and y-axis, respectively. Two functional determinants play important roles here, the Wronskian \[ {\mathcal W}_ n(u)=\det [\partial^{i+j}u/\partial x^ i\partial y^ j]_{i,j=0,...,n} \] and the determinant \[ {\mathcal D}_ n(u)={\mathcal D}_ n(u;x_ 0,...,x_ n;y_ 0,...,y_ n)=\det [u(x_ i,y_ j)]_{i,j=0,...,n}, \] where \((x_ i,y_ j)\) are \((n+1)^ 2\) given points in G. Let us mention some of a series of results: If u is real analytic on G and if on a subdomain of G it has the form (1) with \(A_ k\), \(B_ k\) n-times differentiable, then u has the form (1) (with possibly different \(A_ k\), \(B_ k)\) on the whole domain G. If \(u\in C(G),\) \({\mathcal D}_ n(u)=0\) on G and if in every non-empty open subset \(G'\) of G there exist points \((x_ i,y_ j)\in G',\) \(i,j=0,...,n-1,\) such that \({\mathcal D}_{n-1}(u;x_ 0,...,x_{n-1};y_ 0,...,y_{n- 1})\neq 0,\) then u has the form (1) with continuous \(A_ k\), \(B_ k\). Other assertions concern the convergence of functions (1) in C and \(L^ 2\). Finally, the authors discuss the so called method of extended separation of variables for classical PDEs, which is based on finding solutions of the form (1).
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 35A25
Mathematics Subject Classification ID: 26E05
zbMATH DE Number: 4138076
zbMATH Keywords: sum of products of functions
zbMATH Keywords: partial differential equation
zbMATH Keywords: representation
zbMATH Keywords: Wronskian
zbMATH Keywords: determinant
zbMATH Keywords: method of extended separation of variables
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90031-1
OpenAlex ID: W2080229414
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description: scientific article; zbMATH DE number 4138076