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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