Chapter 0
Background
In the first section below, terms are defined and proofs are included. In the second
section, most terms are defined, but proofs are omitted. In the third section, not
only are proofs omitted, but some terms are undefined. The point is that everything
we do can be made fully rigorous, but neither lack of rigor nor absence of advanced
training in analysis should interfere with the acquisition of Fourier analysis in its
most classical settings.
0.1. Elementary mathematics
An arithmetic progression (sometimes abbreviated AP) is a set of the form {nq+a :
Z}. Hence a sequence {un} is said to be in arithmetic progression if un+1 un is
the same for all n. That means that un = nq + a for some q and a. We frequently
sum such numbers.
Theorem 0.1. If u1,u2,...,uN are consecutive members of an arithmetic progres-
sion, then
(0.1) u1 + u2 + · · · + uN = N ·
u1 + uN
2
.
For example,
1 + 2 + · · · + N =
N(N + 1)
2
.
Proof. Let d be determined so that un+1 un = d for all n. We write the sum
twice, first in its natural order and then in reverse order. Thus if S is the sum, then
S = u1 + u2 + u3 + · · · + uN−1 + uN ,
S = uN + uN−1 + uN−2 + · · · + u2 + u1.
We now sum in columns. On the left hand side we have 2S. In the first column on
the right we have u1 + uN . In the second column on the right we have u2 + uN−1 =
(u1 + d) + (uN d) = u1 + uN . In the third column we have u3 + uN−2 =
1
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