Linear Ill-posed problems are solvable on the average for all gaussian measures

Abstract

The purpose of this paper is to bring a series of rather surprising results which have appeared in the technical literature to the attention of a wider audience. We describe the results informally here; precise definitions and results will be given below. Our subject is the solution of an ill-posed problem on a digital computer. Such a problem can at best only be solved approximately. We say a problem is solvable if we can compute an "-approximation for any positive " and is unsolvable if we cannot compute an "-approximation, even for arbitrarily large ". It was shown in [17] that a linear problem is unsolvable iff it is ill-posed. Are there circumstances under which one can avoid this negative conclusion? It is common to associate an operator S with an ill-posed problem. If S is linear, then a problem is ill-posed iff S is unbounded. Note that unboundedness is a worst case concept; the supremum of S operating on elements in the unit ball of its domain is infinite. This suggests introducing the concept of a problem’s being ill-posed on the average if the expectation of S with respect to a measure is infinite and being well-posed on the average if this expectation is finite. These concepts were introduced in [17], where it was shown that if the measure is Gaussian and the linear operator S is measurable, then a linear problem is unsolvable on the average iff it is ill-posed on the average. A natural next step is to find a linear ill-posed problem which is also ill-posed on the average. Several attempts to do this for Gaussian measures were unsuccessful. This suggested the question: Is every linear problem well-posed on the average for any Gaussian measure? This question was answered in the affirmative in [5] and [15]. The conclusion in the title follows immediately from this and the preceding result. What if the measure is non-Gaussian? Now the situation is more complicated. In particular, there are problems that are even ill-posed on the average. In this paper, we round out the picture by presenting examples of such problems for non-Gaussian measures.