Given a sequence $\varrho =(r_n)_n\in [0,1)$ tending to $1$, we consider the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $\mathbb {D}$ such that for any compact set K included in the unit circle ${\mathbb {T}}$, different from ${\mathbb {T}}$, the set $\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space ${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in $H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${\mathcal {C}}(K)$ for any compact set $K\subset {\mathbb {T}}$ different from ${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:f\mapsto f(r_n \cdot )$ acting on $H(\mathbb {D})$. Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.

Based on the work of Mauldin and Williams [‘On the Hausdorff dimension of some graphs’, Trans. Amer. Math. Soc. 298(2) (1986), 793–803] on convex Lipschitz functions, we prove that fractal interpolation functions belong to the space of convex Lipschitz functions under certain conditions. Using this, we obtain some dimension results for fractal functions. We also give some bounds on the fractal dimension of fractal functions with the help of oscillation spaces.

Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

The systematic development of coarse-grained (CG) models via the Mori–Zwanzig projector operator formalism requires the explicit description of a deterministic drift term, a dissipative memory term and a random fluctuation term. The memory and fluctuating terms are related by the fluctuation–dissipation relation and are more challenging to sample and describe than the drift term due to complex dependence on space and time. This work proposes a rational basis for a Markovian data-driven approach to approximating the memory and fluctuating terms. We assumed a functional form for the memory kernel and under broad regularity hypothesis, we derived bounds for the error committed in replacing the original term with an approximation obtained by its asymptotic expansions. These error bounds depend on the characteristic time scale of the atomistic model, representing the decay of the autocorrelation function of the fluctuating force; and the characteristic time scale of the CG model, representing the decay of the autocorrelation function of the momenta of the beads. Using appropriate parameters to describe these time scales, we provide a quantitative meaning to the observation that the Markovian approximation improves as they separate. We then proceed to show how the leading-order term of such expansion can be identified with the Markovian approximation usually considered in the CG theory. We also show that, while the error of the approximation involving time can be controlled, the Markovian term usually considered in CG simulations may exhibit significant spatial variation. It follows that assuming a spatially constant memory term is an uncontrolled approximation which should be carefully checked. We complement our analysis with an application to the estimation of the memory in the CG model of a one-dimensional Lennard–Jones chain with different masses and interactions, showing that even for such a simple case, a non-negligible spatial dependence for the memory term exists.

We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

We prove that, given $2< p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$, and that, given $1< p<2$, the Fourier series of sequences in $\ell _p$ belong to $L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1< p<\infty$ and every $0\le \alpha <1$, there is a Schauder basis of $\ell _p$ whose conditionality constants grow as $(m^{\alpha })_{m=1}^{\infty }$, and there is an almost greedy basis of $\ell _p$ whose conditionality constants grow as $((\log m)^{\alpha })_{m=2}^{\infty }$.

In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.

For functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$ , $1\notin B$ , then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$ . This answers a question of P. M. Gauthier.

If the logarithmic dimension of a Cantor-type set K is smaller than $1$ , then the Whitney space $\mathcal {E}(K)$ possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in $\mathcal {E}(K)$ can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space $\mathcal {E}(K)$ . We show that these bases are quasi-equivalent.

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form

$$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$

$\omega _{\mathcal {E}}^r(f, t)_p$

$f\in L^p(\Omega )$

$\mathcal {E}\subset \mathbb {S}^{d-1}$

$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

$\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$

$\mathcal {E}$

$\Omega \subset {\mathbb R}^d$

$C^2$

$\Omega \subset {\mathbb R}^d$

$\mathcal {E}$

$\mathcal {N}_d(\Omega )\in {\mathbb N}$

$\mathcal {N}_d(\Omega )$

$\mathcal {E}$

$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

$\Omega $

$r\in {\mathbb N}$

$p>0$

$\mathcal {N}_d(\Omega )=d$

$C^2$

$\Omega \subset {\mathbb R}^d$

$d=2$

$\Omega \subset {\mathbb R}^2$

$d\ge 3$

$\Omega \subset {\mathbb R}^d$

$d\ge 3$

$\Omega \subset {\mathbb R}^d$

$\mathcal {N}_d(\Omega )$

$\Omega $

$\Omega $

$\mathcal {E}\subset \mathbb {S}^{d-1}$

$\mathcal {E}$

$\Omega $

$r\in {\mathbb N}$

$0<p\leq \infty $

$\mathcal {N}_d(\Omega )$

$d\ge 3$

A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body $\Omega \subset {\mathbb R}^d$ , but with the set $\mathcal {E}$ containing more than $(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set:

$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$

$\mathcal {C}[[\mathcal {V}]]$

$L^2([-1,1])$

Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules.

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx. 5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$. We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$ , the identity function ${\mathbf {x}}$ , and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^{\mathbf {x}}}$ . Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $ . As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$ . We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$ , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.

Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $ .

We extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding Nikolskii inequality for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.