Abstract analytic function

analytic mapping of Banach spaces

A function $ f(x) $ acting from some domain $ D $ of a Banach space $ X $ into a Banach space $ Y $ that is differentiable according to Fréchet everywhere in $ D $, i.e. is such that for any point $ a \in D $ there exists a bounded linear operator $ \delta f ( a , \cdot ) $ from $ X $ into $ Y $ for which the following relation is true:

$$ \lim\limits _ {\| h \| \rightarrow 0 } \ \| h \| ^ {-1} \cdot \ \| f ( a + h ) - f ( a ) - \delta f ( a , h ) \| = 0 , $$

where $ \| \cdot \| $ denotes the norm on $ X $ or on $ Y $; $ \delta f ( a , h ) $ is called the Fréchet differential of $ f $ at $ a $.

Another approach to the notion of an abstract analytic function is based on differentiability according to Gâteaux. A function $ f(x) $ from $ D $ into $ Y $ is weakly analytic in $ D $, or differentiable according to Gâteaux in $ D $, if for each continuous linear functional $ y ^ \prime $ on $ Y $ and each element $ h \in X $ the complex function $ y ^ \prime ( f ( x + \xi h ) ) $ is a holomorphic function of the complex variable $ \xi $ in the disc $ | \xi | < \rho ( x , h ) $, where $ \rho ( x, h ) = \sup \{ {| \xi | } : {x + \xi h \in D } \} $. Any abstract analytic function in a domain $ D $ is continuous and weakly analytic in $ D $. The converse proposition is also true, and the continuity condition can be replaced by local boundedness or by continuity according to Baire.

The term "abstract analytic function" is sometimes employed in a narrower sense, when it means a function $ f(z) $ of a complex variable $ z $ with values in a Banach space or even in a locally convex linear topological space $ Y $. In such a case any weakly analytic function $ f(z) $ in a domain $ D $ of the complex plane $ \mathbf C $ is an abstract analytic function. One can also say that a function $ f(z) $ is an abstract analytic function in a domain $ D \subset \mathbf C $ if and only if $ f(z) $ is continuous in $ D $ and if for any simple closed rectifiable contour $ L \subset D $ the integral $ \int _ {L} f(z) dz $ vanishes. For an abstract analytic function $ f(z) $ of a complex variable $ z $ the Cauchy formula (cf. Cauchy integral) is valid.

Let $ f(x) $ be a weakly analytic function in a domain $ D $ of a Banach space $ X $. Then $ f ( x + \xi h ) $, as a function of the complex variable $ \xi $, has derivatives of all orders in the domain $ \widetilde{D} = \{ \xi : {x + \xi h \in D } \} $, $ h \in X $, these derivatives being abstract analytic functions from $ \widetilde{D} $ into $ Y $. If the set $ \{ {x + \xi h } : {| \xi | \leq 1 } \} $ belongs to $ D $, then

$$ f ( x + h ) = \sum _ {n = 0 } ^ \infty \frac{1}{n!} \delta ^ {n} f ( x , h ) , $$

where the series converges in norm, and

$$ \left . \delta ^ {n} f ( x , h ) = \frac{d ^ {n} }{d \xi ^ {n} } f ( x + \xi h ) \right | _ {\xi = 0 } = $$

$$ = \ \frac{1}{2 \pi i } \int\limits _ {| \xi | = 1 } f ( x + \xi h ) \xi ^ {- n - 1 } d \xi . $$

A function $ y = P(x) $ from $ X $ into $ Y $ is called a polynomial with respect to the variable $ x $ of degree at most $ m $ if, for all $ x , h \in X $ and for all complex $ \xi $, one has

$$ P ( x + \xi h ) = \sum _ {\nu = 0 } ^ { m } P _ \nu ( x , h ) \xi ^ \nu , $$

where the functions $ P _ \nu (x, h ) $ are independent of $ \xi $. The degree of $ P(x) $ is exactly $ m $ if $ P _ {m} ( x , h ) \neq 0 $. A power series is a series of the form $ \sum _ {n=0} ^ \infty P _ {n} (x) $ where $ P _ {n} (x) $ are homogeneous polynomials of degree $ n $ so that $ P _ {n} ( \alpha x ) = \alpha ^ {n} P _ {n} ( x ) $, $ x \in X $, for all complex $ \alpha $. An arbitrary weakly convergent power series $ \sum _ {n=0} ^ \infty P _ {n} (x) $ in a domain $ D $ converges in norm towards some weakly analytic function $ f(x) $ in $ D $, and $ P _ {n} (x) = \delta ^ {n} f(0, x)/n! $, $ 0 \in D $. A function $ f(x) $ is an abstract analytic function if and only if it can be developed in a power series in a neighbourhood of all points $ a \in D $

$$ f ( a + h ) = \sum _ {n = 0 } ^ \infty P _ {n} ( h ) , $$

where all $ P _ {n} (h) $ are continuous in $ X $.

Many fundamental results in the classical theory of analytic functions — such as the maximum-modulus principle, the uniqueness theorems, the Vitali theorem, the Liouville theorem, etc. — are applicable to abstract analytic functions if suitable changes are introduced. The set of all analytic functions in a domain $ D $ forms a linear space.

The notion of an abstract analytic function can be generalized to wider classes of spaces $ X $ and $ Y $, such as locally convex topological spaces, Banach spaces over an arbitrary complete valuation field, etc.

References

[1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2]	R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
[3]	L. Schwartz, "Cours d'analyse" , 2 , Hermann (1967)