corresponding to the r zero eigenvalues. Then:
Lemma 2. Under Assumption 3 and the conditions of Lemma 1,
■ ^ï
RrTD (1) Y1 ^F1(x)2dx
Fk(1)RrτD,Z
RrM(!) ft
RM z (Fk) n
V k 7
jointly in k = 1,...,m, where the Yk ’s andZ are independent q-variate standard normally
distributed, with Yk defined by (13). Moreover, Z does not depend on Fk .
Such weight functions Fk do exist. In particular,
Lemma 3. If Fk(x) = cos(2kπx), then the conditions (6) through (10) hold. Moreover, we then
have Fk(1) = 1, ʃʃFk(x)Fk(y)min(xy) dxdy = 1(kπ y2, ʃFk(x)2dx = 1.
There are many ways to choose these functions Fk, but as will be shown in section 5, the
above choice is optimal in some sense.
Denoting
Fk(x )2 dx
γ k = j
y ^^I^k(x )Fk((y )min(x, y ) dxdy
, δk
Fk (1)
↑∣ ʃFk (x )2 dx
(14)
it follows now easily from Lemmas 1-2:
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