Sums of products of functions of x times functions of y
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).
method of extended separation of variables
scientific article; zbMATH DE number 4138076
scientific article