Optimal Estimation of Linear Operators in Hilbert Spaces from Inaccurate Data

Among all possible methods of estimating a linear operator U from inaccurate data, smoothing of the data followed by U is an optimal procedure. This result which we formulate and prove here in a Hilbert space setting may be viewed as an extension of the Golomb–Weinberger method of estimation. It provides as well a rationale for an optimal choice of a smoothing parameter. Our results also provide a method of finding the exact value of the n-width of subsets of a Hilbert space determined by two ellipsoidal bounds. We give an application of this method to the space of essentially time- and band-limited signals introduced by D. Slepian.