Using Integral Transforms to Estimate Higher Order Derivatives

Using Integral Transforms to Estimate Higher Order Derivatives

When doing error analysis for numerical quadrature, achieving good uniform bounds on higher order derivatives of the integrand is paramount. As undergraduates become increasingly adept with programmable calculators, numerical integration schemes such as Simpson’s Rule and the Trapezoidal Rule take on a new relevance. Although it may be a rare calculus class that dwells overmuch on error bounds ...

Fuzzy transforms of higher order approximate derivatives: A theorem

In many practical applications, it is useful to represent a function f(x) by its fuzzy transform, i.e., by the “average” values Fi = ∫ f(x) ·Ai(x) dx ∫ Ai(x) dx over different elements of a fuzzy partition A1(x), . . . , An(x) (for which Ai(x) ≥ 0 and n ∑ i=1 Ai(x) = 1). It is known that when we increase the number n of the partition elements Ai(x), the resulting approximation get closer and cl...

Integral Transforms of Functions with Restricted Derivatives

First- and Higher-Order Correlation Detection Using Wavelet Transforms

In order to detect intermittent firstand higher-order correlation between a pair of signals in both time and frequency, a wavelet-based coherence and bicoherence technique was developed. Due to the limited averaging in a time-frequency coherence estimate, spurious correlated pockets were detected due to statistical variance. The introduction of multiresolution, localized integration windows was...

Local image analysis using higher-order Riesz transforms

The Reisz transform is a mathematical operator that has been the subject of recent image analysis research. It can be used to model local image structure as a superposition of sinusoids, where phase describes feature type (line or edge) and amplitude describes feature strength. The advantage of this split of identity is that local symmetry can be analysed separately from overall intensity. In t...