Georgia Institute of Technology College of Sciences

Sandro Levi and I have investigated variational strengthenings of uniform continuity and uniform convergence of nets or sequences of functions with respect to a family of subsets of the domain. Out of our theory comes an answer to this basic question: what is the weakest topology stronger than the topology of pointwise convergence in which continuity is preserved under taking limits? We argue that the classical theory constitues a misunderstanding of what is fundamentally a variational phenomenon.

In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant Laplace operators acting on Lie groups of polynomialgrowth and elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.

In a regression model, say Y_i=f(X_i)+\epsilon_i, where (X_i,Y_i) are observed and f is an unknown regression function, the errors \epsilon_i may satisfy what we call the "weak'' assumption that they are orthogonal with mean 0 and the same variance, and often the further ``strong'' assumption that they are i.i.d. N(0,\sigma^2) for some \sigma\geq 0. In this talk, I will focus on the polynomial regression model, namely f(x) = \sum_{i=0}^n a_i x^i for unknown parameters a_i, under the strong assumption on the errors. When a_i's are estimated via least squares (equivalent to maximum likelihood) by \hat a_i, we then get the {\it residuals} \hat epsilon_j := Y_j-\sum_{i=0}^n\hat a_iX_j^i. We would like to test the hypothesis that the nth order polynomial regression holds with \epsilon_j i.i.d. N(0,\sigma^2) while the alternative can simply be the negation or be more specific, e.g., polynomial regression with order higher than n. I will talk about two possible tests, first the rather well known turning point test, and second a possibly new "convexity point test.'' Here the errors \epsilon_j are unobserved but for large enough n, if the model holds, \hat a_i will be close enough to the true a_i so that \hat epsilon_j will have approximately the properties of \epsilon_j. The turning point test would be applicable either by this approximation or in case one can evaluate the distribution of the turning point statistic for residuals. The "convexity point test'' for which the test statistic is actually the same whether applied to the errors \epsilon_j or the residuals \hat epsilon_j avoids the approximation required in applying the turning point test to residuals. On the other hand the null distribution of the convexity point statistic depends on the assumption of i.i.d. normal (not only continuously distributed) errors.