A2 Partition of Unity

Let ç (x) be a continuous function and let us define s(x)=$3 Ç(x _ k) Vxe [0,1].

Clearly, the resulting function s is periodic and satisfies s(x) = s(x + 1). Thus, under wide conditions, its Fourier series can be expressed as s(x) = £ Snei2™x n e Z

with

By substitution into the Fourier series, we have that s(x) = Z 9(2ftn)e'2nnx, neZ

from which it is possible to deduce the equivalence between s(x) = 1 and ç(2nn) = ôn. Note that a rigorous demonstration of this equivalence requires that s(x) converge uniformly.

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