Smoothing of bump functions

Smoothing of Bump Functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C∞-smooth bump which depends locally on finitely many coordinates.

Smoothing by Spline Functions

Definable Smoothing of Continuous Functions

Let R be an o-minimal expansion of a real closed field. Given definable continuous functions f : U → R and : U → (0,+∞), where U is an open subset of Rn, we construct a definable Cm-function g : U → R with |g(x)− f(x)| < (x) for all x ∈ U . Moreover, we show that if f is uniformly continuous, then g can also chosen to be uniformly continuous.

Image Smoothing with Exponential Functions

Noise reduction in images, also known as image smoothing, is an essential and rst step before further processings are done on the image. The key to image smoothing is to preserve important features while removing noise from the image. Gaussian function is widely used in image smoothing. Recently it has been reported that exponential functions (value of the exponent is not equal to 2) perform su...

Smoothing Spline Estimation of Variance Functions

This article considers spline smoothing of variance functions. We focus on selection of smoothing parameters and develop three direct data-driven methods: unbiased risk (UBR), generalized approximate cross validation (GACV) and generalized maximum likelihood (GML). In addition to guaranteed convergence, simulations show that these direct methods perform better than existing indirect UBR, genera...