# Infinite Product Representations of Kernel Functions on Fractals

## Document Type

Presentation Abstract

## Presentation Date

3-6-2017

## Abstract

A fractal is, loosely speaking, an image which exhibits a degree of self-similarity. In the early eighties Benoit Mandelbrot published the classic, Fractal Geometry of Nature, in which he argued that many natural phenomena were better modeled by fractals. The fractals we will consider are obtained through the iteration of quadratics, although the results presented here hold in much greater generality.

Shortly after Mandelbrot's publication, mathematicians began looking for ways to extend various branches of analysis to fractals. Among these branches is functional analysis, and in particular we will look at a recently developed method for constructing a kernel function on a given fractal.

Uniquely associated to any kernel function is a Hilbert space of functions in which every linear evaluation functional is bounded. These spaces are called reproducing kernel Hilbert spaces, and as a Hilbert space, notions of length and angle can be defined. The construction we follow represents the kernel function as an infinite product involving iterations of the quadratic defining the fractal of interest. We will determine precisely which quadratics this construction holds for.

## Recommended Citation

Tipton, James, "Infinite Product Representations of Kernel Functions on Fractals" (2017). *Colloquia of the Department of Mathematical Sciences*. 524.

https://scholarworks.umt.edu/mathcolloquia/524

## Additional Details

Monday, March 6, 2017 at 3:00 p.m. in Math 103

Refreshments at 4:00 p.m. in Math Lounge 109