Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

The minimum is realized for
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 645
xk = bk
n
ak
k=1
For any subset I ⊂ N let us denote as before by 〈fn | n ∈ I〉 the closure of the linear space
generated by the set of vectors (fn | n ∈ I) in a Hilbert space H .
We note that the distance d(fn+1;〈f1,...,fn〉) of the vector fn+1 in H from the hyperplane
〈f1,...,fn〉 may be calculated in terms of the Gram determinants Γ(f1,f2,...,fk) corresponding
to the set of vectors f1,f2,...,fk (see [10]):
d fn+1;〈f1,...,fn〉 = min
t=(tk)∈R n
fn+1 +
b 2 k
ak
n
k=1
−1
.
tkfk
2
= Γ(f1,f2,...,fn+1)
, (6)
Γ(f1,f2,...,fn)
where the Gram determinant is defined by Γ(f1,f2,...,fn) = det γ(f1,f2,...,fn) and
γ(f1,f2,...,fn) =: γn is the Gram matrix
Lemma 10. We have
⎛
(f1,f1) (f1,f2) ...
⎞
(f1,fn)
⎜ (f2,f1)
γ(f1,f2,...,fn) = ⎜
⎝
(f2,f2) ...
. ..
(f2,fn) ⎟
⎠
(fn,f1) (fn,f2) ... (fn,fn)
.
d fn+1;〈f1,...,fn〉 =
det γn+1
det γn
where dn+1 = ((f1,fn+1), (f2,fn+1),...,(fn,fn+1)) ∈ R n .
Proof. We may write
n
k=1
tkfk − fn+1
2
=
n
k,m=1
tktm(fk,fk) − 2
= (fn+1,fn+1) − γ −1
n dn+1,dn+1
,
n
k=1
= (γnt,t)− 2(t, dn+1) + (fn+1,fn+1),
tk(fk,fn+1) + (fn+1,fn+1)
where t = (t1,t2,...,tn) ∈ Rn . Using (58) for An = γn we get
(γnt,t)− 2(t, dn+1) = γn(t − t0), (t − t0) − γ −1
n dn+1,dn+1
,
where t0 = γ −1
n dn. Hence we get (see (6))
min
t=(tk)∈R n
fn+1 −
n
k=1
tkfk
2
= min
t=(tk)∈Rn
(γnt,t)− 2(t, dn+1) + (fn+1,fn+1)