Expository Notes

Expository Notes on Introduction to Harmonic Analysis

1. (f̂ + g)(ξ) = f̂(ξ) + ĝ(ξ). 2. For any α ∈ C, (α̂f)(ξ) = αf̂(ξ). 3. Let f̄ denote the complex conjugate of f . Then ˆ̄ f(ξ) = f̂(ξ). 4. For λ ∈ R \ {0}, denote fλ(x) = λf(λx). Then f̂λ(ξ) = f̂( ξ λ ). 5. For y ∈ R let (τyf)(t) = f(t− y). Then τ̂yf(ξ) = f̂(ξ)e−2πiξy. 6. |f̂(ξ)| ≤ ‖f‖L1(R). Definition 1.3. The Schwartz space, S(R), is the space of C∞ functions that are rapidly decreasing, in the sense that

Expository Notes on Signal Processing on Graphs

Graph theory has developed into a useful tool in applied mathematics as many modern data sets are or can be represented as a graph. In these data sets, vertices correspond to different sensors, observations, or data points and edges represent connections, similarities, or correlations among those points. Some immediate applied examples include social network data, electricity networks, and imag...

Luxury Prices: an Expository Note*

Economists generally associate the redistribution of resources with the apparatus of taxes and transfer payments. Such redistributions are done by the power of the authorities. However, resources are redistributed by other means as well. People give away income in a variety of ways, deliberate and unintentional. In this paper, agents transfer consumption goods in return for a good which lacks m...

SEGMENTATION OF EXPOSITORY TEXT Marti

This paper describes TextTiling, an algorithm for partitioning expository texts into coherent multi-paragraph discourse units which re ect the subtopic structure of the texts. The algorithm uses domain-independent lexical frequency and distribution information to recognize the interactions of multiple simultaneous themes. Two fully-implemented versions of the algorithmare described and shown to...

The Vigenére Cipher Expository Paper