We present a new approach to the construction of biorthogonal wavelet transforms using polynomial splines. The construction is performed in a “lifting” manner and we use interpolatory, as well as local quasi-interpolatory and smoothing splines as predicting aggregates in this scheme. The transforms contain some scalar control parameters which enable their flexible tuning in either time or frequ...

In this document I develop a weight function theory of zero order basis function interpolants and smoothers. In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples of ...

A new algorithm is proposed for demodulating discrete-time AM-FM signals, which first interpolates the signals with smooth splines and then uses the continuous-time energy separation algorithm (ESA) based on the Teager-Kaiser energy operator. This spline-based ESA retains the excellent time resolution of the ESAs based on discrete energy operators but performs better in the presence of noise. F...

A frequency interpretation of Bayesian "confidence" intervals for smoothing splines. Abstract The frequency properties of Wahba's Bayesian "confidence" intervals for smoothing splines are investigated by a large sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals is averaged across the observation points, we explain why the av...

The asymptotic behavior of penalised spline estimators is studied in the univariate case. B-splines are used and a penalty is placed on mth order differences of the coefficients. The number of knots is assumed to converge to ∞ as the sample size increases. We show that penalised splines behave similarly to Nadaraya-Watson kernel estimators with an “equivalent” kernels depending upon m. The equi...

I discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thinplate spline smoothing problem, and are optimal in the sense that the truncation is designed to result in the minimum possible pe...