1.7. Linear equations in primes 115
and the claim follows.
Exercise 1.7.2. Show that the estimate (1.54) for q 2 can only hold when
ν is bounded uniformly in N; this explains the presence of the hypothesis
q 2 in the above condition.
Exercise 1.7.3. Show that the estimate (1.54) is equivalent to the estimate
En∈Z/NZ|
ξ∈Z/NZ
g(ξ)e(ξnx/N)|ν(n) C g
q (Z/NZ)
for all g : Z/NZ C, where q := q/(q 1) is the dual exponent to q.
Informally, this asserts that a Fourier series with
q
coefficients can be “re-
stricted” to the support of ν in a uniformly absolutely integrable manner
(relative to ν). Historically, this is the origin of the term “restriction theo-
rem” (in the context where Z/NZ is replaced with a Euclidean space such
as
Rn,
and ν is a surface measure on a manifold such as the sphere
Sn−1).
See for instance [Ta2003].
Now we turn to the approximation property. The approximation g to
f needs to be close in
u2
norm, i.e., the Fourier coefficients need to be
uniformly close. One attempt to accomplish this is hard thresholding: one
simply discards all Fourier coefficients in the Fourier expansion
f(n) =
ξ∈Z/NZ
ˆ(ξ)e(xξ/N)
f
of f that are too small, thus setting g equal to something like
g(n) =
ξ∈Z/NZ:|
ˆ(ξ)|≥ε
f
ˆ(ξ)e(xξ/N).
f
The main problem with this choice is that there is no guarantee that the
non-negativity of f will transfer over to the non-negativity of g; also, there
is no particular reason why g would be bounded.
But a small modification of this idea does work, as follows. Let S :=
Z/NZ : |
ˆ(ξ)|
f ε} denote the large Fourier coefficients of f. The
function g proposed above can be viewed as a convolution f K, where
K(n) :=

ξ∈S
e(xξ/N) and f K(n) := Em∈Z/NZf(m)K(n m). The
inability to get good pointwise bounds on f K can be traced back to the
oscillatory nature of the convolution kernel K (which can be viewed as a
generalised Dirichlet kernel).
But experience with Fourier analysis tells us that the behaviour of such
convolutions improves if one replaces the Dirichlet-type kernels with some-
thing more like a Fej´ er-type kernel instead. With that in mind, we try
g(n) := Em1,m2∈Bf(n + m1 m2)
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