Allan P. Donsig

Convexity and Optimization

Optimization is a central theme of applied mathematics that involves minimizing or maximizing various quantities. This is an important application of the derivative tests in calculus. In addition to the first and second derivative tests of one-variable calculus, there is the powerful technique of Lagrange multipliers in several variables. This chapter is concerned with analogues of these tests that are applicable to functions that are not differentiable. Of course, some different hypothesis must replace differentiability, and this is the notion of convexity. It turns out that many applications in economics, business, and related areas involve convex functions. As in other chapters of this book, we concentrate on the theoretical underpinnings of the subject. The important aspect of constructing algorithms to carry out our program is not addressed. However, the reader will be well placed to read that material. Results from both linear algebra and calculus appear regularly.

Approximation by Polynomials

This chapter introduces some of the essentials of approximation theory, in particular approximating functions by “nice” ones such as polynomials. In general, the intention of approximation theory is to replace some complicated function with a new function, one that is easier to work with, at the price of some (hopefully small) difference between the two functions. The new function is called an approximation. There are two crucial issues in using an approximation: first, how much simpler is the approximation? and second, how close is the approximation to the original function? Deciding which approximation to use requires an analysis of the trade-off between these two issues.

Limits of Functions

There are several reasonable definitions for the limit of a sequence of functions. Clearly the entries of the sequence should approximate the limit function f to greater and greater accuracy in some sense. But there are different ways of measuring the accuracy of an approximation, depending on the problem. Different approximation schemes generally correspond to different norms, although not all convergence criteria come from a norm. In this section, we consider two natural choices and see why the stronger notion is better for many purposes.

Discrete Dynamical Systems

Suppose we wish to describe some physical system. The dynamical systems approach considers the space X of all possible states of the system—think of a point x in X as representing physical data. We will assume that X is a subset of some normed vector space, often ({mathbb{R}}). The evolution of the system over time determines a function T of X into itself that takes each state to a new state, one unit of time later.

Norms and Inner Products

In this chapter, we generalize to more abstract settings two key properties of ({mathbb{R}^n}): the Euclidean norm of a vector and the dot product of two vectors. The generalizations, norms and inner products, respectively, are set in a general vector space. Many of our applications will be set in this framework.

Differentiation and Integration

In this chapter, we examine the mathematical foundations of differentiation and integration. The theorems of this chapter are useful not only to make calculus work but also for studying functions in many other contexts.We do not spend any time on the important applications that typically appear in courses devoted to calculus, such as optimization problems. Rather we will highlight those aspects that either depend on or apply to results in real analysis.

Fourier Series and Physics

Fourier series were first developed to solve partial differential equations that arise in physical problems, such as heat flow and vibration. We will look at the physics problem of heat flow to see how Fourier series arise and why they are useful. Then we will proceed with the solution, which leads to a lot of very interesting mathematics. We will also see that the problem of a vibrating string leads to a different PDE that requires similar techniques to solve.

The Real Numbers

Doing analysis in a rigorous way starts with understanding the properties of the real numbers. Readers will be familiar, in some sense, with the real numbers from studying calculus. A completely rigorous development of the real numbers requires checking many details. We attempt to justify one definition of the real numbers without carrying out the proofs.

Fourier Series and Approximation

A natural problem is to take a wave output and decompose it into its harmonic parts. Engineers are able to do this with an oscilloscope. A real difficultly occurs when we try to put the parts back together. Mathematically, this amounts to summing up the series obtained from decomposing the original wave. In this chapter, we examine this delicate question: Under what conditions does a Fourier series converge?