On the optimal approximation rates for criss-cross finite element spaces

Let Δ denote the triangulation of the plane obtained by multi-integer translates of the four lines x=0, y=0, x=y and x=?y. By $\text{l}{}_{\mathrm{k,\; h}}{}^{\mathrm{\mu }}$ we mean the space of all piecewise polynomials of degree ?k with respect to the scaled triangulation hΔ having continuous partial derivatives of order $\text{?μ}\text{on}\text{R}{}^2$. We show that the approximation properties of $\text{l}{}_{\mathrm{k,\; h}}{}^{\mathrm{\mu }}$ are completely governed by those of the space spanned by the translates of all so called box splines contained in $\text{l}{}_{\mathrm{k,h}}{}^{\mathrm{\mu }}$. Combining this fact with Fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where μ is the largest degree of smoothness for which these spaces are dense as h tends to zero. Furthermore, we study the question of local linear dependence of the translates of the box splines for the above criss-cross triangulations.     

Cardinal hermite interpolation with box splines

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.     

Multivariate splines and hyperplane arrangements

In this paper, we use the so-called conformality method of smoothing cofactor (abbr. CSC) and hyperplane arrangements to study truncated powers and box splines in R². By the relation between hyperplane arrangements and truncated powers, we give the expressions of the truncated powers. Moreover, by means of the CSC method, the least smoothness degrees of the truncated powers and the box splines on each direction of partition edges are studied.     

Exponential box splines

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZ ^s and a vector of constants λ ?R ⁿ, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofR ⁿ—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).     

Multivariate F-splines and Fractional Box Splines

We introduce multivariate F-splines, including multivariate F-truncated powers T _f (?|M) and F-box splines B _f (?|M). The classical multivariate polynomial splines and multivariate E-splines can be considered as a special case of multivariate F-splines. We document the main properties of T _f (?|M) and B _f (?|M). Using T _f (?|M), we extend fractional B-splines to fractional box splines and show that these functions satisfy most of the properties of the traditional box splines. Our work unifies and generalizes results due to Dahmen-Micchelli, de Boor-Höllig, Ron and Unser-Blu, and also presents a new tool for computing the integration over polytopes.     

Basis of graded identities of the superalgebra M 1,2(F)

Denote by Mat _k,l (F) the algebraM _n (F) of matrices of order n = k + l with the grading (Mat _k,l ⁰ (F),Mat _k,l ¹ (F)), where Mat _k,l ⁰ (F) admits the basis $$ \{ e_{ij} ,i \leqslant k,j \leqslant k\} \cup \{ e_{ij} ,i > k,j > k\} $$ and Mat _k,l ¹ (F) admits the basis $$ \{ e_{ij} ,i \leqslant k,j > k\} \cup \{ e_{ij} ,i > k,j \geqslant k\} . $$ . Denote byM _k,l (F) the Grassmann envelope of the superalgebra Mat _k,l (F). In the paper, bases of the graded identities of the superalgebras Mat_1,2(F) and M _1,2(F) are described.     

Shape-preserving interpolation by splines using vector subdivision

We give a local convexity preserving interpolation scheme using parametricC ² cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

A system of functions

A system of functions $$f_k (x) = \sum\nolimits_{i = 1}^r a _i \varphi _\iota (x)^k + b_i \overline {\varphi _\iota } (x)^k , k = 1,2,...$$ is considered on the interval [0,l]. Under certain conditions on the? _i(x), it is proved that the system 1 ∪ {f_k(x)} _k=1 ^∞ is complete in the space L_p(0,l). In the case r=1 it is proved, under certain additional assumptions, that the system {f_k(x)} _k=0 ^∞ is minimal.     

Quantization of the action of U(k,l) on R2(K + l)

The natural action of U(k, l) on $\text{C}$^{k + l} leaves invariant a real skew non-degenerate bilinear form B, which turns $\text{C}$^{k + l} into a symplectic manifold (M, ω). The polarization F of M defined by the complex structure of $\text{C}$^{k + l} is non-positive. If L is the prequantization complex line bundle carried by (M, ω), then U(k, l) acts on the space $\text{U}$ of square-integrable $\text{L ? ΛF}{}^1$ forms on M, leaving invariant the natural non-degenerate, but non-definite, inner product ((·, ·)) on $\text{U}$. The polarization F also defines a closed, densely defined covariant differential ?? on $\text{U}$ which is U(k, l)-invariant. Let $\text{⊥}$ denote orthocomplementation with respect to ((·, ·)). It is shown that the restriction of ((·, ·)) to the U(k, l)-stable subspace ? (Ker ??) ∩ (Im ??)^$\text{⊥}$ is semi-definite and that the unitary representation of Uk, l on the Hilbert space $\text{H}$ arising from ? by dividing out null vectors is unitarily equivalent to the representation of U(k, l) obtained from the tensor product of the metap ectic and $\text{Det}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ representations of MU(k, l), the double cover of U(k, l).     

Generalized balancing numbers

The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1^k + 2^k + … + (x — 1)^k = (x + 1)^l + … + (y — 1)^l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.     

A note on bivariate box splines on a k-direction mesh

We determine the dimension of the polynomial subspace of the linear space spanned by the translates over lattice points of a bivariate box spline on a k-direction mesh.     

Robust Hurwitz stability via sign-definite decomposition

In this paper, we first consider the problem of determining the robust positivity of a real function f(x) as the real vector x varies over a box X∈R^l. We show that, it is sufficient to check a finite number of specially constructed points. This is accomplished by using some results on sign-definite decomposition. We then apply this result to determine the robust Hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest. We develop an eight polynomial vertex stability test that is a sufficient condition of Hurwitz stability of the family. This test reduces to Kharitonov’s well known result for the special case where the parameters are just the polynomial coefficients. In this case, the result is tight. This test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set. The result is illustrated by examples.     

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order $k\geqslant 2$ spanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval $[x_i,x_{i+1}\rangle\subset [0,1]$ , i?=?0,1, ...l. Most of the paper deals with non-polynomial case m _i ,n _i ?∈?[4,?∞?), and polynomial splines known as VDP–splines are the special case when m _i , n _i are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator $\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}$ . Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.     

Bivariate box splines and smooth pp functions on a three direction mesh

Let S denote the space of bivariate piecewise polynomial functions of degree ? k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For $\text{ρ = ?}\text{(2k ? 2)}\text{3}\text{?}$, k ? 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for $\text{S}{}_{\mathrm{<mtext>loc</mtext>}}\text{= {f ∈ S:}\text{supp}\text{f ? Ω}}$, with Ω a convex subset of $\text{R}$².     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Forbidding just one intersection

Following a conjecture of P. Erdös, we show that if $\text{F}$ is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ? 2l + 2 and n sufficiently large |$\text{F}$| ? (_{k ? l ? 1}^{n ? l ? 1}) with equality holding if and only if $\text{F}$ consists of all the k-sets containing a fixed (l + 1)-set. In general we show |$\text{F}$| ? d_kn^{max;{;l,k ? l ? 1};}, where d_k is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3).     

Extreme points for positive forms on lp

We characterize, in the finite-dimensional case, the extreme points of the convex set of bilinear forms on l^p x l^q (p > 1, q > 1, p^-1 + q^-1⩽1) which have nonnegative coefficients and norm bounded by one. The results are extended to the multilinear case.     

A two-sided omega-theorem for an asymmetric divisor problem

In this article we study the arithmetic functiond _a,b (l,k;n) which is defined as the number of representationsn=v ^a w ^b withw lying in the residue classl modulok (a,b andl,k fixed positive integers). For the remainder term in the asymptotic formula for Σ_n≤xd_a,b(l,k,;n) we obtain an Ω_± (under a certain restriction onl andk) which is sharper than the known results for the corresponding “unrestricted” problem.