Encoding and recovery of operator values (Q1190535)

Let \(X\) be a metric space and let \(M_ n=\{\mu_ 1,\mu_ 2,\dots,\mu_ N\}\) be a set of continuous functions \(\mu_ k\) on \(X\). Then for every element \(x\in X\) there is assigned an information vector \(T(x,M_ N)=\{\mu_ 1(x),\mu_ 2(x),\dots,\mu_ N(x)\}\), i.e., a point in \(R^ N\). The mapping \(x\to T(x,M_ N)\) is called an encoding of element \(x\) and the set \(M_ N\) defines a method of encoding. The author considers the more general problem of recovering the values of the operator \(y=Ax\) with the help of discrete information, namely the vector \(T(x,M_ N)\), which is taken from the element \(x\), where \(A\) is an operator from \(X\) into \(Y\). The operator \(A\) may be considered as known, or it may be allowed to be given implicitly. For example, the operator \(A\) may define the solution of the equation \(By=x\), where \(B\) is a given operator from \(Y\) into \(X\). The effectiveness of the encoding method is measured by the diameter of the set \(\{y:\;y=Az,z\in X,\;T(z,M_ N)=T(x,M_ N)\}\). Concrete situations are considered where \(A\) is the \(m^{th}\) differentiation operator or the convolution operator, and also where \(A\) is the solution operator of a boundary-value problem.