Let X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

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.

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.

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.

We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.

In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

We define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

A generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

In this paper we deal with the operators

$$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*}$$

${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] $

${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.

Necessary and sufficient conditions for the incompleteness of exponential system in Cα are characterised, where Cα is the weighted Banach space of complex continuous functions f defined on ℝn with f(t)exp(−α(t)) vanishing at infinity in the uniform norm.

We propose a collocation method for solving initial value problems of second-order ODEs by using modified Laguerre functions. This new process provides global numerical solutions. Numerical results demonstrate the efficiency of the proposed algorithm.

We introduce a family of orthogonal functions, termed as generalized Slepian functions (GSFs), closely related to the time-frequency concentration problem on a unit disk in D. Slepian [19]. These functions form a complete orthogonal system in with , and can be viewed as a generalization of the Jacobi polynomials with parameter (α, 0). We present various analytic and asymptotic properties of GSFs, and study spectral approximations by such functions.

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in ${{\mathbf{R}}^{n}}$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by “analytic” and “meromorphic” solutions of the equation $Lu\,=\,0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

For we show that the conditions for all open sets G, C2,q denoting Bessel capacity, are not sufficient to characterize the compact sets X with the property that each function harmonic on and in Lp(X) is the limit in the Lp norm of a sequence of functions which are harmonic on neighbourhoods of X.

We characterize open subsets U of ℝN in which the bounded solutions of certain elliptic equations can be approximated pointwise by uniformly bounded solutions that are continuous in Ū. This result is established in terms of certain capacities. For closed subsets X, this characterization allows us to approximate bounded solutions in X° uniformly on relatively closed subsets of X° by solutions continuous on certain subsets of the boundary of X.

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.