Results & Lemmas (9)
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THEOREM 1.
THEOREM 1. Suppose that (2.1)- (2.4) hold and let h be a complex sequence. Then the following are equivalent: (i) Σΐ=oa ( k h)k converges;…
THEOREM 1. Suppose that (2.1)- (2.4) hold and let h be a complex sequence. Then the following are equivalent: (i) Σΐ=oa ( k h)k converges; (ii) h belongs to the domain of B (iii) Σΐ=ohkpk(z) converges for all z in some uniqueness set for Ψuψir
THEOREM 2.
THEOREM 2. Suppose that (2.1)-(2.4) hold and let h and y be complex sequences. Then y = Bh if and only if Σ^=o hkpk(z) is uniformly…
THEOREM 2. Suppose that (2.1)-(2.4) hold and let h and y be complex sequences. Then y = Bh if and only if Σ^=o hkpk(z) is uniformly convergent on compact subsets of Ω to the function f whose power series at 0 is given by The problem of explicitly determining those sequences y which
THEOREM 3.
THEOREM 3. Suppose that (2.1)-(2.4) hold. A complex sequence h belongs to 3€ if and only if each of the series (2.5) is convergent. We note…
THEOREM 3. Suppose that (2.1)-(2.4) hold. A complex sequence h belongs to 3€ if and only if each of the series (2.5) is convergent. We note here for later use that if (2.5) converges, then (2.6)
THEOREM 4.
THEOREM 4. Suppose in addition to (2.1)-(2.4) that (2.7) X hkpk (z) = 0 for all z<ΞΩ implies k=0 that hk = 0, k = 0,1,2,. For each f E 3*…
THEOREM 4. Suppose in addition to (2.1)-(2.4) that (2.7) X hkpk (z) = 0 for all z<ΞΩ implies k=0 that hk = 0, k = 0,1,2, . For each f E 3* with set 11/11 = sup Then & is a Banach space, {pk}o is a basis for SF9 and the mapping k=n { 30 "V oo Σ ak(λh)k\
LEMMA 1.
LEMMA 1. Suppose that uk l is a sequence of d x d invertible matrices such that (3.1) Σ ||Wfc+i-M*||σ
LEMMA 1. Suppose that {uk}l is a sequence of d x d invertible matrices such that (3.1) Σ ||Wfc+i-M*||σ
LEMMA 2.
LEMMA 2. Let E be a uniqueness set forfuf2,
LEMMA 2. Let E be a uniqueness set forfuf2,
LEMMA 3.
LEMMA 3. Suppose that (2.1) —(2.3) hold and let h be a complex sequence such that Σΐ=oak(λh)k converges. Then (3.2) J hkPk(z) k=0 converges…
LEMMA 3. Suppose that (2.1) —(2.3) hold and let h be a complex sequence such that Σΐ=oak(λh)k converges. Then (3.2) J hkPk(z) k=0 converges if and only if (3.3) J Pk(z)(λh)k /c=0 converges. Moreover, (3.2) converges uniformly on compact subsets of ίl if and only if the same is true of (3.3).
THEOREM 5.
THEOREM 5. Suppose that (2.1)-(2.4) hold and let h be a complex sequence. Then the following are equivalent: (i) ΣΓ=0 ak (λh )k converges…
THEOREM 5. Suppose that (2.1)-(2.4) hold and let h be a complex sequence. Then the following are equivalent: (i) ΣΓ=0 ak (λh )k converges (ii) ΣΓ=0 Pΐ(0)(λh )k converges for j = 0,1,2, , (iii) Σk c=0Pk(z)(λh)k converges for all z in some uniqueness set for (Pi,(P2,--,<Pdl (iv) Σk=0Pk(z)(λh)k
Lemma 2
Lemma 2 and (2.4), choose points zu z2,, zd in Ω such that the matrix (<Pi(zk))91 ^ /, k ^ d, is invertible. In proving that (i) implies…
Lemma 2 and (2.4), choose points zu z2, , zd in Ω such that the matrix (<Pi(zk))91 ^ /, k ^ d, is invertible. In proving that (i) implies (iii) we can take {zl9 ,z«f} as our uniqueness set; for the reverse implication we will suppose that zί9 ,zd are chosen from some arbitrary uniqueness set.