We propose a mathematical limit of L1-stable weak asymptotic methods. A family of L1-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex ...domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.
We study boundary values of holomorphic functions in translation-invariant distribution spaces of type
. New edge of the wedge theorems are obtained. The results are then applied to represent
as a ...quotient space of holomorphic functions. We also give representations of elements of
via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations and heat kernel representations in the context of the Schwartz spaces
,
, and their weighted versions.
A differential geometric method is introduced to study the modulus of boundary values of holomorphic functions on smoothly bounded pseudoconvex domains $D$ in $\mathbf{C}^n, n \geqslant 2$. It is ...shown that functions in $A(D)$ are determined up to a constant factor by their modulus on an open subset of the Shilov boundary. For the case of $H^\infty(D)$, it is shown that inner functions which satisfy a certain local condition are constant.
We consider a bounded, circular, strictly convex domain
Ω
with
C
2
boundary. We show that there exists a holomorphic function
f
1
, continuous to the boundary such that every slice function has a ...series of Taylor coefficients divergent with every power
p
∈
0
,
2
)
. We also construct inner function
f
2
which every slice is also inner and has series of Taylor coefficients with the same property. Next we generalize it to obtain
f
3
∈
O
(
Ω
)
with given modulus a.e. on all slices and Taylor series as above.
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