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Sorry for the impreciseness of the title. It is merely meant for an analogy.

Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For instance, let $f: \mathbb{R}^2 \to \mathbb{R}$ be Borel measuable. Let $f(\cdot, u) \in C^k(I)$ for some open set $I$ and for all $u$ in a Borel set $D$. Let

$g = \int_D f(x,u) {\rm d} u$.

Then a sufficient condition for $g \in C^k(I)$ is that $f^{(k)}(x, \cdot)$ is dominated by an integrable function on $D$, i.e., $\sup_{x \in I} |f^{(k)}(x, \cdot)| \in L^1(D)$, and $g^{(k)}(x) = \int_D f^{(k)}(x,u) {\rm d} u$ holds in $I$.

My question is about when is real-analyticity preserved under integration, say, if $f$ is real-analytic in $I$ for each $u$, i.e., $f(\cdot, u) \in C^{\omega}(I)$ for all $u \in D$, what will be a sufficient condition for $g \in C^{\omega}(I)$?

Following the above rationale, we will obtain the following condition: for each $x_0 \in I$,
1) the radius of convergence of $f(x, u) = \sum_k a_k(u) (x-x_0)^k$ is bounded away from zero for all $u \in D$.
2) integrability condition: $\int_D \sum_k a_k(u) (x-x_0)^k {\rm d} u < \infty$. Then the analyticity of $g$ follows from Fubini's theorem.

Questions:
1) Is there other sufficient condition different from the above 'superficial' generalization, maybe exploring other characterization of real analyticitiy? The absolute integrability might not be easy to check.
2) Is there a more local version, which might give the radius of convergence of $g$.

Thanks!

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1 Answer 1

up vote 5 down vote accepted

It is rather hard to work with the coefficients of the Taylor expansion directly but, fortunately, one does not have to. Real analyticity is actually analiticity in a strip (of varying width) around the real line and it is quite hard to imagine the situation when $f(x)$ can be computed/estimated on the line but not nearby. So, for all "practical" purposes, it is enough to check that for every point $x\in\mathbb R$, there is a disk $B\subset\mathbb C$ centered at $x$ such that the integrals $\int_D|f(z,u)|\,du$ are uniformly bounded on $B$. Then the analyticity follows from Fubini and Cauchy formula.

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