Once upon a time, students of mathematics and students of science or engineering took the same courses in mathematical analysis that went beyond calculus. A choice between encountering Fourier series as a small example of the representation theory of Banach algebras, and encountering Fourier series on their own and developed in an ad hoc way, is no choice at all. On the other hand, the practitioner of mathematics still needs a variety of clear tools and has not had the time to acquire the broadest and most definitive knowledge: learning necessary and sufficient conditions when simple sufficient conditions can serve that purpose, or to learn the necessary and sufficient conditions. general framework covering several examples.
First, the ideas and methods of the theory of distributions lead to formulations of classical theories which are mathematically satisfactory and complete, and which at the same time provide the most useful perspective for applications. As an example, the basic L2 theory of Fourier series can be established quickly and without reference to measure theory once L2(0, 27r) is known to be complete. Here L2(0, 21r) is viewed as a subspace of the space of periodic distributions and shown to be a Hilbert space.
It is easy to derive necessary and sufficient conditions that a formal trigonometric series is the Fourier series of a distribution, an L2 distribution, or a smooth function. I am indebted to Max Jodeit and Paul Sally for lively discussions of what and how analysts should learn, to Nancy for her support throughout, and especially to Fred Flowers for his excellent handling of the manuscript.
Basic Concepts
Ifx,yeP, then x + yeP
The axioms given so far do not yet determine 118; all of them are satisfied by the subset Q of rational numbers. Similarly, if there is an x e R such that every y E A satisfies x < y, then A is said to be bounded below and x is called a lower bound for A. Similarly, x is a greater bound lower for A if it is a lower bound and if every other lower bound x' satisfies x' <- x.
If A is a nonempty subset of Fl bounded above, then A has a smallest upper bound. So x - h is an upper bound for A less than x, and x is not the smallest upper bound. The argument that proves Theorem 2.1 proves the following: A is upward bounded; the smallest upper limit x is positive; if x2 < 2 then x would not be an upper bound, while if x2 > 2 then x would not be the smallest upper bound.
Consistency of axioms and existence of 112; can be proved (to the satisfaction of most mathematicians) by constructing 118 starting with the rational values. It can be shown that any set with addition, multiplication and subset of positive elements satisfying all the above axioms is just a copy of R.
For anyz,weC,
The number R defined in the statement of Theorem 4.8 is called the radius of convergence of the power series (4.2). It is the radius of the largest circle in the complex plane within which (4.2) converges. The following is an analogy for the closed sets of Proposition 5.1. a) Suppose that z is a limit point of the set C.
A subsequence of this sequence is a sequence of the form (yk)k 1, where for each k there is a positive integer nk such that. Thus, (yk)k 1 is just a selection of some (perhaps all) xn's, taken in order. Let S be a nonempty subset of the vector space X and let Y be the set of all linear combinations of elements of S.
If Y is a subspace of the finite dimensional vector space X, then there is a subspace Z with the properties Y n Z = {0}; dim Y + dim Z = dim X;. As in the proof of Theorem 2.1, the limit (3.1) exists if and only if the limits of the real and imaginary parts of this expression exist. Prove the following extension of the mean value theorem: if f and g are continuous real-valued functions on [a, b] and if the derivatives exist at every point of (a, b), then there is c e (a, b) ) that.
This means that the coefficients of the power series (4.6) are uniquely determined by the function f (provided that the radius of convergence is positive). Trigonometric functions and the logarithm 57. have the form a,x°, where this series converges for all x. As a function from (0, oo) to R, the logarithm is thus the inverse of the exponential function.
We show that there is no way to choose a norm on 9 such that convergence as defined above is equivalent to convergence in the sense of the metric associated with the norm. However, there is a way to choose a metric on 9 (not associated with a norm) so that convergence in the sense of 9 is equivalent to convergence in the sense of the metric. A set of functions (u )n 1 c -0 is a Cauchy set in the sense of 9 if and only if it is a Cauchy set in the sense of the metric d'.
There is an important generalization of the Banach space concept that includes spaces such as 9 Let X be a vector space over real or complex numbers. Therefore, any function of the form (1) can be written as a polynomial in the trigonometric functions cos x and sin x.
In this section we consider two infinite-dimensional analogues of the finite-dimensional complex of the CN Hilbert space. The proof of this theorem is very similar to the proof of Theorem 3.1. d) the set of linear combinations of em elements is dense in 1+2;. In this case a are called the Fourier coefficients of the function u and the formal series.
If u e ' and u are continuously differentiable, then the partial sums of the Fourier series of u converge uniformly to u. Extend the result of Exercise 3 to the partial sums of the Fourier series of a distribution F e L2. An important feature of such a program is to express the action of F e 9' on u e 9 in terms of the respective series of Fourier coefficients.
We want to calculate the Fourier coefficients of the convolutions in terms of the Fourier coefficients F, G and u. Our formulation of the problem for the distribution is as follows: Given G E Y', find the distributions Ft e j9' for each t > 0 such that. It is easy to see that the maximum value of v is reached at one of the edges in question.
A second proof of the uniform convergence of u to g as t -+ 0 is sketched in the following exercises. Imagine the rod divided into sections of length e, and assume that x is the coordinate of the middle of a section. As in the case of the heat equation, we begin by formulating a corresponding problem for periodic distributions and solving it.
The force on the string at point x is due to the string's tension. Also, let's assume that the length of the string is 7rl instead of in. Formula (15) shows that the pitch varies inversely with length and radius and directly with the square root of the voltage.
Use Exercise 2 and the results of this section to prove a theorem about the existence and uniqueness of classical solutions to the problem. Then u is the unique solution of the Dirichlet problem in the unit disk with g as the value on the boundary.
Complex Analysis
Then, using Cauchy's integral theorem and the argument given in the proof of Theorem 3.1, we can replace C with Cl in (8). Verify the Cauchy integral formula of the form (1) by direct calculation when f(z) = e$ and C is a circle. Show that any function that is meromorphic in the entire plane and is holomorphic in co or has a pole there is a rational function. exists and is a holomorphic function z for Re z > 0.
Therefore, (5) defines the function in the disc if it is only assumed that g is a periodic distribution. Condition (10) is not necessarily satisfied by coefficients of the power series (9) converging in the disc. Thus, condition (10) gives a subset of the set of all holomorphic functions in the disk.
