By Rockswold G. K.

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B) In particular, if B is a C ∗ -subalgebra of A, then ∂S M (B) = M (B) and so Mβ (A) = ∅ for each β ∈ M (B). Via α1 ∼ α2 ⇐⇒ α1 |B = α2 |B (α1 , α2 ∈ M (A)) an equivalence relation is given on M (A). The corresponding partition of M (A) into equivalence classes is nothing else than the partition of M (A) into ﬁbers over M (B). This set of equivalence classes equipped with a natural topology is homeomorphic to M (B). (c) Now suppose Y is a compact Hausdorﬀ space and B is a C ∗ -subalgebra of C(Y ) containing the constant functions.

The conjugate function. If F is a real-valued harmonic function in D, then there is a real-valued harmonic function G in D such that F + iG is analytic in D. Any two such functions G diﬀer only by a real constant and the function G for which G(0) = 0 is called the conjugate function of F and denoted by F . If F is a complex-valued harmonic function in D, then F can be written as F = U + iV , where U and V are real-valued, and F is deﬁned as U + iV . Now let f ∈ L1 . One can show that the conjugate function of the harmonic extension hf possesses non-tangential limits almost everywhere on T.

Moreover, H ∞ and CA are Banach algebras. H 2 is a Hilbert space and we have f ∈ H 2 if and only if n∈Z+ |fn |2 < ∞; the set {χn }n∈Z+ forms an orthogonal basis in H 2 . 40. The analytic extension. If f ∈ H 1 , then the harmonic extension f is an analytic function in D and it is therefore also referred to as the analytic extension. Thus, for f ∈ H 1 one has fn z n f (z) = n≥0 (z ∈ D). 38(b). Fatou’s theorem can be made more precise as follows. (a) Let F be an analytic function in D, put Fr (eiθ ) = F (reiθ ), and suppose that sup Fr p < ∞, where 1 ≤ p ≤ ∞.

### A0-stable linear multistep formulas of the-type by Rockswold G. K.

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