From now on our analysis will concentrate on real-valued conditionally positive de¬nite

functions that possess a positive generalized Fourier transform. Then by Corollary 8.13 we

can express a typical quadratic form via

2

N N

’d/2 iω T x j

± j ±k (x j ’ xk ) = (2π ) ±je (ω)dω.

Rd

j,k=1 j=1

(ω) ≥

Thus an appropriate is given if its Fourier transform satis¬es (ω) and is of

order less than or equal to the order of .

Lemma 12.2 Let χ M be the characteristic function of B(0, M), M > 0, i.e. χ M (x) = 1 if

x 2 ¤ M and χ M (x) = 0 otherwise. Then

’d/2

χ M (x) = (χ M )∨ (x) = M d/2 x Jd/2 (M x 2 ),

2

where Jν is a Bessel function of the ¬rst kind.

Proof As χ M is a radial function, its Fourier transform and inverse Fourier transform

coincide and can be computed via Theorem 5.26 as

M

’(d’2)/2

χ M (x) = x t d/2 J(d’2)/2 ( x 2 t)dt.

2

0

210 Stability

Using the de¬nition of Jν , the dominated convergence theorem and the multiplication

property of the -function give

M∞

(’1)m ( x 2 t/2)2m+(d’2)/2 d/2

’(d’2)/2

χ M (x) = x t dt

m! (m + d/2)

2

0 m=0

∞

(’1)m ( x 2 /2)2m+(d’2)/2 M

’d/2+1

=x t 2m+d’1 dt

m! (m + d/2)

2

0

m=0

∞

(’1)m ( x 2 /2)2m+(d’2)/2 1

’d/2+1

=x M 2m+d

m! (m + d/2) d + 2m

2

m=0

’d/2

= M d/2 x Jd/2 (M x 2 ).

2

This function is the key ingredient in ¬nding the function .

Theorem 12.3 Let be an even conditionally positive de¬nite function that possesses a

positive Fourier transform ∈ C(Rd \ {0}). With the function

•0 (M) := inf (ω)

ω 2 ¤2M

a lower bound on »min is given by

d

•0 (M) M

»min (A ,X ) ≥

2 (d/2 + 1) 23/2

for any M > 0 satisfying

1/(d+1)

π (d/2 + 1)

2

12

M≥ (12.2)

qX 9

or, a fortiori,

6.38d

M≥ . (12.3)

qX

Proof Let us de¬ne by its Fourier transform as

•0 (M) (d/2 + 1)

(ω) ≡ (χ M — χ M )(ω),

M (ω) :=

2d M d π d/2

where f — g denotes the convolution from Theorem 5.16. Then ≥ 0 has support in

B(0, 2M), which shows that (ω) ≥ (ω) for ω 2 > 2M. For ω 2 ¤ 2M note that

•0 (M) (d/2 + 1)

(ω) ¤ vol(B(0, 2M)) ¤ •0 (M) ¤ (ω).

2d M d π d/2

12.2 Lower bounds for »min 211

This shows that is a good candidate and that we have to bound the quadratic form for .

This is done directly. First note that

•0 (M) (d/2 + 1)

(χ M — χ M )∨ (x)

=

M (x) d M d π d/2

2

•0 (M) (d/2 + 1)

= (2π)d/2 |χ M (x)|2

d M d π d/2

2

•0 (M) (d/2 + 1)

x ’d Jd/2 (M x 2 ).

= 2

2

2d/2

Next we use

N

± j ±k ’ xk ) ≥ ± ’ |± j ||±k || ’ xk )|

2

M (x j M (0) M (x j

2

j=k

j,k=1

1

≥± ’ (|± j |2 + |±k |2 )| ’ xk )|

2

M (0) M (x j

2

2 j=k

⎛ ⎞

N

⎝ ’ xk )|⎠ .

≥± ’ max |

2

M (0) M (x j

2

1¤ j¤N k=1

k= j

By Proposition 5.6 we know that

d

•0 (M) M

= .

M (0)

(d/2 + 1) 23/2

Hence, the stated bound on »min is M (0)/2 and it remains to show that

N