Entire functions in Bo-algebras

Composition in ultradifferentiable classes

“…Since all M λ are weakly log-convex, E (M) (R) is a Fréchet algebra which is locally m-convex, by [29], i.e., E (M) (R) has an equivalent seminorm system {p} such that p(f g) ≤ p(f )p(g) for all f, g ∈ E (M) (R). So for each λ ∈ Λ, compact K ⊆ R, and ρ > 0 there exist p, µ ∈ Λ, compact L ⊆ R, σ > 0 and constants C, D > 0 such that…”

Superposition in Classes of Ultradifferentiable Functions

“…Let us assume that the Fréchet algebra E (ω) (R; C) is holomorphically closed. Then, by [20], E (ω) (R; C) is a locally m-convex algebra. Therefore we find a continuous multiplicative seminorm q, positive constants C, B, a and k ∈ N such that for each f ∈ E (ω) (R; C) and each m ∈ N,…”

“…According to a theorem of Mitiagin, Zelazko and Rolewicz [20] (see also [12]), a Fréchet algebra A (over the field K of real or complex numbers) is locally m-convex if, and only if, for every a ∈ A and for every entire function φ(z) = ∞ n=0 c n z n (with coefficients c n ∈ K), the series ∞ n=0 c n a n converges in A. The next argument is taken from [9].…”

Superbosonization via Riesz Superdistributions

“…If γ • S (µ) = ∅, then µ ∈ S (V, V¯0 ,R ), in the sense that µ extends continuously to S (V, V¯0 ,R ). DEFINITION C. 36. Let E be a test space for (V, V¯0 ,R ) and µ ∈ E , where we assume γ • S (µ) = ∅.…”