On metric theorems in interpolation theory (Q1901826)

Notations. Let \(f(z)\) be a square-integrable function defined on a set \(\mathbb{X}\) of unit \(\mu\)-measure. Let \(\mu^k\) denote the \(k\)-dimensional measure on the space \((x_1,\dots,x_k)\), \(x_i\in \mathbb{X}\) induces by \(\mu\). The interpolation problem investigated is: Given linearly independent functions \(\phi_1,\dots,\phi_n\in\mathbb{L}^2_\mu\) find constants \(c_1,\dots,c_n\) such that for suitably chosen \(x_j\in\mathbb{X}\), \(j=1,\dots,n\) the equations \[ \sum^n_{i=1} c_i\phi(x_j)= f(x_j)\tag{1} \] hold. A slightly more general problem is also mentioned whereas equation (1) is substituted by \[ \mathbb{L}_j\Biggl[\sum^n_{i=1} c_i\phi_i\Biggr] (x_j)= \mathbb{L}_j[f](x_j)\tag{2} \] for some operators \(\mathbb{L}_j\) defined on a subset of \(\mathbb{L}^2_\mu\). Obviously the existence of an interpolation depends on the non-vanishing of the determinant \(|\phi_i(x_j)|^n_{i,j=1}\). With this goal in mind the author proves the following three theorems. Theorem 1. If a measure \(\mu\) is such that the functions \(\phi_1,\dots,\phi_n\) are orthogonizable then \[ \mu^n\{\text{det}|\phi_i(x_j)|^n_{i,j=1}\neq 0\}>0. \] In fact \[ \int\mu^n(dQ)\text{det}|\phi_i(x_j)|^n_{i,j=1}= n! \prod^n_{i=1} \int\phi^2_i(x)(dx), \] where \(Q\) is the set on which the above determinant does not vanish. Theorem 2. For any orthogonal system \(\phi_1,\dots,\phi_n\) the inequality \[ \mu^{n-m}\{x_{m+1},\dots,x_n\mid |\phi_i(x_j)|^n_{i,j=1}\neq 0\}>0 \] holds for the given \(x_1,\dots,x_m\) if and only if there exists a set of integers \(1\leq j_1<j_2<\cdots< j_m\leq n\) such that the determinant \[ |\phi_{j_1},\dots,\phi_{j_m}(x_j)|^n_{i,j=1}\neq 0. \] As a particular case of the problem summarized in equation (2), the author considers the following situation. Let \(\mathbb{X}\) be a compact subset of \(\mathbb{R}^s\), \(x_j=(x^{(1)}_j,\dots,x^{(s)}_j)\). Assume that \(f(x)\) has partials with respect to a certain \(x^{(\ell)}\) for almost all \(x\in\mathbb{X}\). What are the conditions under which the measure of the set \((x_1,\dots,x_{n-1})\) for which \((\text{mod }\mu^{n-1})\) there exists a unique solution to the system \[ \sum_ic_i\phi_i(x_j)= f(x_j)(\text{mod }\mu),\quad j=1,\dots,n-1,\tag{3} \] \[ \Biggl[\sum_i c_i\phi_i\Biggr]'(x_{n-1})= [f]'(x_{n-1}, (\text{mod }\mu)) \] is positive. Prime denotes differentiation with respect to \(x^{(\ell)}\). An answer is given in theorem 3. The inequality \[ \mu^{n-1}\{x_1,\dots,x_{n-1}\mid \Delta(Q)\neq 0\}>0 \] holds, where \[ \Delta(Q)= \text{det}|\phi_k(x_1),\dots,\phi_k(x_{n-1}),\;\phi'(x_{n-1})|^n_{k=1}, \] if and only if at least one pair of functions \(\phi_k\), \(\phi_\ell\), \(k\neq\ell\) is linearly independent on a subset of positive \(\mu\)-measure, where at least one of these functions differs from zero. The proof uses the Cauchy-Binet theorem. In the proof the equation \(\phi_1(x)=\text{const}\cdot \phi_2(x)\) should be written as \(\phi_1(x)\neq\text{const}\cdot\phi_2(x)\). The author notes that the method of least squares \[ \int\lambda(dx)\Biggl[\sum^n_{i=1} c_i\phi_i(x)-f(x)\Biggr]^2=\min \] for a finite measure \(\lambda\) on \(\mathbb{X}\) falls under the above method. Indeed the corresponding determinant of the normal linear system \[ \text{det}|\int \phi_i(x)\phi_j(x)\lambda(dx)|^n_{i,j=1} \] equals to \[ {1\over n!} \int \lambda(dx_1)\cdots\int \lambda(dx_n)(\text{det}|\phi_i(x_j)|)^2 \] and is positive if \(\lambda\) is absolutely continuous with respect to \(\mu\) and if the conditions of theorem 1 hold.