By Paul A. Baran
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Example text
O) have the same limit. • 6 Suppos e that (in) is a Cauchy sequence in 11. 0 converging pointwise to i and that limn-t oo < i n' i n >1lo = 0 (U«) tends to 0 in the norm sense). Then i == o. LE MMA 17 Theory Proof. 'r/x E E, I(x) = lim In(x) n-t oo lim ex(fn) = 0 by assumption c). = n-too Thus we can define an inner product on 1i by setting • < I, g >1l= n---too lim < In' s« »u; where (In)(resp. (gn)) is a Cauchy sequence in 110 converging pointwise to I (resp. g). ,. ,. >1lo has those properties.
E2 (X ), with suitable index set. e2 (X ) is the set of complex sequences {x cn 0:' EX} satisfying L 2 Ix a l < 00 a EX endowed with the inner product < (xa), (Ya) >= L xaYa' a EX Theorem 4 provides a characterization of ALL reproducing kernels on an ab stract set E. e2 (X ) (see Fortet , 1995) . At first sight this characterization can appear mainly theoretical. It is not the case. Indeed t his theorem provides an effective way of constructing reproducing kernels or of proving that a given function is a reproducing kernel.
Y))](x) where II V denotes the orthogonal projection onto the space V . Proof. As II V is a self-adjoint operator (II V = IIir), by Theorem 9 K V is the kernel of II v. Now, the restriction of II V to the su bspace V is the identity of V. Therefore Kv is the reproducing kernel of V. • The Riesz's representation theorem guarantees that any continuous linear functional u 1i -+ f t------t C u (J) 30 RKHS IN PROBABILITY AND STATISTICS has a representer u in 11. , < I, U >= u(j). 7) In the case of a RKHS, U can be expressed easily through the kernel as stated in the following lemma.