We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb {R}_{\mathcal {G}}$ and the reduct of $\mathbb {R}_{\text {an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the gamma function on $(0,\infty )$ and the zeta function on $(1,\infty )$.

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the $o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

Let ${{C}^{M}}$ denote a Denjoy–Carleman class of ${{C}^{\infty }}$ functions (for a given logarithmically-convex sequence $M\,=\,\left( {{M}_{n}} \right))$. We construct: (1) a function in ${{C}^{M}}\left( \left( -1,\,1 \right) \right)$ that is nowhere in any smaller class; (2) a function on $\mathbb{R}$ that is formally ${{C}^{M}}$ at every point, but not in ${{C}^{M}}\left( \mathbb{R} \right)$; (3) (under the assumption of quasianalyticity) a smooth function on ${{\mathbb{R}}^{p}}\,\left( p\,\ge \,2 \right)$ that is ${{C}^{M}}$ on every ${{C}^{M}}$ curve, but not in ${{C}^{M}}\left( {{\mathbb{R}}^{p}} \right)$.

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated.