Abstract: Let ϕ be a linear combination of certain box splines and\(\hat \phi \) its Fourier transform, such that\(\hat \phi \left( 0 \right) e 0\) and\(D^\beta \hat \phi \left( {2\pi k} \right) = 0\) for all κ∈ZN{0} and β≤α. In this paper we construct an expression of the multivariate polynomial (·-y)α in terms of a linear combination of the integer translates of ϕ(·), where the coefficients can be computed recursively using only the information on\(D^\beta \hat \phi \left( 0 \right)\), β ≤ α. As an application, a quasi-interpolation scheme based only on function values on (scaled) integers κ∈ZN is constructed that gives a “multivariate order” of approximation that includes both coordinate and total orders.

Abstract: We introduce a definition of free multivariate splines which generalizes the univariate notion of splines with free knots. We then concentrate on the simplest case, piecewise constant functions and characterize some classes of functions which have a prescribed order of approximation inLp by these splines. These characterizations involve the classical Besov spaces.

Abstract: A new proof is given thatn distinct points on the unit circle can be mapped inton arbitrary points on the unit circle of the complex plane by a finite Blaschke product A result of this proof is that the mapping can be done with at mostn−1 factors in the product The problem is studied in the context of its application to frequency transformations used to design digital filters

Abstract: Given a formal power seriesf(z)≔∑j=0∞ajzj for which the quantityaj−1aj+1/aj2 has a prescribed asymptotic behavior asj→∞, we obtain the asymptotic behavior of poles of rows of the Pade table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag-Leffler functions) and forn=1,2,3,..., that the poles of [m/n](z) approach ∞ with rateam/am+1 asm→∞.

Abstract: This paper is concerned with some efficient algorithms for the calculation of triangular splines. Their development is based on some different interpretations of a construction given by Malcolm Sabin in 1977 [Sabin 1977].

Abstract: For an example where the functions have different branch points we derive the asymptotics of diagonal Hermite-Pade polynomials of type I. The method uses an integral equation obtained by approximating a reproducing kernel. The results are consistent with a new conjecture on the asymptotics of the polynomials associated with more general functions with different branch points.

Abstract: We study the rate of uniform approximation by Norlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions\(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and\(\tilde f^{(1,1)} (x,y)\).

Abstract: The differential correction algorithm for generalized rational functions is described, and two theorems on convergence and order of convergence are given. An example shows that the order of convergence may deteriorate from superlinear to linear when a best generalized rational approximation does not exist.

Abstract: A necessary and a sufficient alternation condition for strongly unique best spline approximations with free knots is given. In the case of simple knots these conditions coincide, and strongly unique best approximations and strongly unique local best approximations are the same. The numerical consequences are discussed.

Abstract: This paper introduces the idea of cardinal interpolation on submodules of Zd by translates of box splines if the condition of global linear independence fails to hold. In particular, the special case of the 4-direction box splines is discussed, where the pertinent submodule is given by the pairs (k, l) of integersk, l withk+l even. For this case, one obtains results that parallel the known results for the 3-direction box splines.

Abstract: A definition is given of the “variation” of a surface which generalizes those considered previously. It is then shown that the variation of a Bernstein polynomial on a triangle is bounded by that of its Bezier net and conditions are derived under which the bound is attained. A bound is also given for the variation of the Bezier net in terms of the variation of the function itself. Finally, it is mentioned how these results lead to variation diminishing properties of certain approximation operators involving polyhedral splines.

Abstract: The central objective of this paper is to discuss linear independence of translates of discrete box splines which we introduced earlier as a device for the fast computation of multivariate splines. The results obtained here allow us to draw conclusions about the structure of such discrete splines which have, in particular, applications to counting the number of nonnegative integer solutions of linear diophantine equations.

Abstract: TheLevy radius for a set of probability measures satisfying certain standard moment conditions is introduced, through the Levy distance of these measures from the unit measure at a fixed point of the real line. Using a moment optimal result of Selberg, an algebraic algorithm is given for the exact calculation of this radius.

Abstract: We discuss the relationship between divided differences, fundamental functions of hyperbolic equations, multivariate interpolation, and polyhedral splines.

Abstract: The bivariate Bernstein-Schoenberg operatorVT of degreem, introduced in [5], is a spline approximation operator that generalizes the Bernstein polynomial operatorBm. It is shown here that for a convex functionf,f≤VT(f)≤Bm(f). This result is then used to show that for a twice differentiable functiong, the asymptotic error limm(VT(g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances thatVT is, in a sense, asymptotically close toBm.

Abstract: The study of cardinal interpolation (CIP) by the span of the lattice translates of a box spline has met with limited success. Only the case of interpolation with the box spline determined by the three directionsd 1=(1, 0),d 2=(0, 1), andd 3=(1, 1) inR 2 has been treated in full generality [2]. In the case ofR d,d ≥ 3, the directions that define the box spline must satisfy a certain determinant condition [6], [9]. If the directions occur with even multiplicities, then this condition is also sufficient. For Hermite interpolation (CHIP) both even multiplicities and the determinant condition for the directions does not prevent the linear dependence of the basis functions. This leads to singularities in the characteristic multiplier when using the standard Fourier transform method. In the case of derivatives in one direction, these singularities can be removed and a set of fundamental splines can be given. This gives the existence of a solution to CHIP inL p (R d) for data inl p (Z d), 1≤p≤2.

Abstract: Letf∈A ρ (ρ>1), whereAρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δl,n−1(f; z) (see (2.3)) has been studied extensively. Recently, V. Totik has obtained some quantitative estimates for\(\overline {\lim _{n \to \infty } } \max _{\left| z \right| = R} \left| {\Delta _{l,n - 1}^ - \left( {f;z} \right)} \right|^{1/n} \). Here we investigate the order of pointwise convergence (or divergence) of Δl,n−1(f; z), i.e., we study\(B_1 \left( {f;z} \right) = \overline {\lim _{n \to \infty } } \left| {\Delta _{l,n - 1} \left( {f;z} \right)} \right|^{1/n} \). We also study some problems arising from the results of Totik.

Abstract: LetP(N,m;r 1,...,r n ) be the class of 1-periodic perfect splines of degreem with 2N knots, which haven distinct zeros in one period with multiplicitiesr 1,...,r n , respectively. We show that there exists a unique extremal elementP *∈P(N,m;r 1,...,r n ) of minimal uniform norm which equioscillates. This problem is related to the optimal recovery of smooth periodic functions on the basis of the Hermitian data.

Abstract: LetEn(f) denote the sup-norm-distance (with respect to the interval [−1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likeex, cosx, etc., the order ofEn(f) asn→∞ is well known. A typical result is $$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$ It is shown in this paper that 2n−1n!En−1(ex) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJk(x)) the quantity $$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$ possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).

Abstract: Let {on(dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szego's theory is concerned with the asymptotic behavior ofon(dμ) when logμ′∈L1. In what follows we will discuss the asymptotic behavior of the ratioon(dμ2)/on(dμ1) on the unit circle whendμ1 anddμ2 are close in a sense (e.g.,dμ2=gdμ1, where g≥0 is such thatQ(eit)g(t) andQ(eit)/g(t) are bounded for a suitable polynomialQ) and μ1′>0 almost everywhere or (a somewhat weaker requirement) limn→∞Φn(dμ1,0)=0 for the monic polynomial Φn. The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.

Abstract: We prove some new relations between functions defined as shadows of cones (cone splines) and simplices (simplex splines). We use them to show how ans-variate simplex spline of some orderk can be written as a sum ofk+1 (s-l)-variate simplex splines of orderk-1. A recurrence relation on the spatial dimension of the simplex spline,s, is proposed as an interesting alternative to the recurrence relation in [17], where one uses the orderk for recursion, but not the spatial dimensions.

Abstract: We consider the problem of C1 interpolation to data given at the vertices and mid-edge points of a tessellation in Rn. The given data are positional and gradient information at the vertices, together with the gradient at the mid-edge points. By subdividing eachn-simplex in an appropriate way, we show how to solve the interpolation problem using piecewise cubic polynomials. The subdivision process is the key to the method and is inductive in nature. It is systematically built up from the two-dimensional case where a variant of the well-known Clough-Tocher element is used.

Abstract: The Caratheodory-Fejer method provides a way for the construction of near best rational approximations. Gutknecht and Trefethen [7], [10] observed that in many cases the constructed functions yield very good estimates of the degree of rational approximation. We will Show that the correct asymptotic is obtained when the method is applied to Stieltjes functions. This fact is of interest in connection with Magnus' considerations of the 1/9-conjecture [9].

Abstract: Generalizing previous work [2], we study complex polynomials {π k },π k (z)=z k +⋯, orthogonal with respect to a complex-valued inner product (f,g)=∫ 0 π f(e iθ)g(e iθ)w(e iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ 0 π w(e iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π k } in the case of the Jacobi weight functionw(z)=(1−z)α(1+z)β, α>−1, and its special case $$\alpha = \beta = \lambda - \tfrac{1}{2}$$ (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ n (z). It has regular singular points atz=1, −1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.