Smoothing causal 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

On sub-ideal causal smoothing filters

Smoothing causal linear time-invariant filters are studied for continuous time processes. The paper suggests a family of causal filters with almost exponential damping of the energy on the higher frequencies. These filters are sub-ideal meaning that a faster decay of the frequency response would lead to the loss of causality.

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...