Uniform analyticity of orthogonal projections

Abstract:

Let $X$ denote the circle $T$ or the interval $[ - 1,1]$, and let $d\mu$ denote a nonnegative, absolutely continuous measure on $X$ . Under what conditions does the Gram-Schmidt procedure in the weighted space ${L^2}(X,{\omega ^2}\;d\mu )$ depend analytically on the logarithm of the weight function $\omega$? In this paper, we show that, in numerous examples of interest, $\log \omega \in BMO$ is a sufficient (often necessary!) condition for analyticity of the Gram-Schmidt procedure. These results are then applied to establish the local analyticity of certain infinite-dimensional Toda flows.