On the recurrence relations for B-splines defined by certain L-splines

This paper provides a general unified recurrence relation for a class of B-splines defined by certain constant coefficient operators. The associated relations for polynomial, trigonometric, hyperbolic, and some particular Tchebycheffian B-splines are all covered as special cases.     

Wr,p(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension $\text{F}{}_1$ of a function $\text{?:E → R}$ (i.e., the H^r,p-spline with knots in E) and studies the cone $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ of all such splines. We study the problem of determining when $\text{F}{}_1$ is in W^r,p ≡ H^r,p ∩ L^p. If $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$, then $\text{F}{}_1$ is called a W^r,p-spline, and we denote by $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ the cone of all such splines. If E is quasiuniform, then $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$ if and only if . The cone $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ with E quasiuniform is shown to be homeomorphic to l^p. Similarly, $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size $\text{¦ E ¦}$ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of $\text{¦ E ¦}$ are derived.     

Shape preserving interpolation using quadratic X-splines

In this note, we use a new approach to define the Quadratic X-splines and then examine it for preserving shape when applied to strictly convex data. Here, the construction of the Quadratic X-Spline interpolation becomes straight forward.     

Topological singularities in W S,P ( S N , S 1)

We are interested in the location of the singularities of maps u ∈ W ^s,p (S ^N , S ¹) when 1 ≤ sp and 1 < sp < 2. To this end, we consider the distributional Jacobian. We show that the range of this operator on W ^s,p (S ^N , S ¹) is the closure in W ^s−2,p ⋂ W ^−1,sp of the set of N − 2-currents defined as the integration on smooth oriented N − 2-dimensional boundaryless submanifolds.     

W-algebras related to parafermion algebras

We study a W-algebra of central charge 2(k−1)/(k+2), k=2,3,…, contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k,0) of type with level k. We calculate the operator product expansions of the W-algebra. We also calculate some singular vectors in the case k6 and determine the irreducible modules and Zhu's algebra. Furthermore, the rationality and the C₂-cofiniteness are verified for such k.     

Approximation by linear combinations of multivariate B-splines

This paper is concerned with the approximation of functions by linear combinations of multivariate B-splines. We construct and analyze local linear approximation schemes on certain uniform configurations. Furthermore we point out how these uniform configurations may be refined locally while still preserving the desired global smoothness of the splines.     

Conformal biderivations of loop W (a,b) Lie conformal algebra

We study conformal biderivations of a Lie conformal algebra. First, we give the definition of a conformal biderivation. Next, we determine the conformal biderivations of loop W(a, b) Lie conformal algebra, loop Virasoro Lie conformal algebra, and Virasoro Lie conformal algebra. Especially, all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.     

Frobenius manifolds from regular classical W-algebras

We obtain polynomial Frobenius manifolds from classical W-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.     

M0(G)-boundaries are M(G)-boundaries

Let M(G) denote the convolution algebra of finite regular complex-valued Borel measures on a locally compact abelian group G, and let M₀(G) be the ideal consisting of those measures whose Fourier-Stieltjes transforms vanish at infinity. Then there is a natural inclusion of the maximal ideal space Δ₀ of M₀(G) in the maximal ideal space of M(G). The main result states that any subset of Δ₀ which is a boundary for M₀(G) is a boundary for M(G). An immediate corollary is that the ?ilov boundary of M₀(G) is dense in the ?ilov boundary of M(G).     

The Luzin approximation of functions from classes Wαp on metric spaces with measure

In this paper we prove an analog of the Luzin theorem on correction for spaces of the Sobolev type on an arbitrary metric space with a measure, satisfying the doubling condition. The correcting function belongs to the H?lder class and approximates a given function in the metrics of the initial space. Dimensions of exceptional sets are evaluated in terms of Hausdorff capacities and volumes. Original Russian Text ? V.G. Krotov and M.A. Prokhorovich, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 55–66. Dedicated to the memory of Petr Lavrent’evich Ul’yanov     

The W-convexity and normal structure in banach spaces

Let X be a Banach space, S(X) - x ε X : #x02016; = 1 be the unit sphere of X.The parameter, modulus of W^*-convexity, W^*(ε) = inf <(x − y)/2, f_x> : x, y S(X), x − y ≥ ε, f_x Δ_x , where 0 ≤ ε ≤ 2 and Δ_x S(X^*) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W^*(ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W^*(1 + ε) > ε/2 where W^*(1 + ε) = lim_→ε W^* (1 + ), then X has normal structure.     

Average Width and Optimal Interpolation of the Sobolev-Wiener Class Wpq(R) in the Metric Lq(R)

In this paper, we determine the exact value of average n − K width _n(W^r_pq(R), L_q(R)) of Sobolev-Wiener class W^r_pq(R) in the metric L_q(R) for 1 > q ≥ p > ∞ and get the value of _n(W^r_p(R), L_qp(R)) for the dual case. We also solve the optimal interpolation problems of W^r_pq(R) in the metric L_q(R) and W^r_p(R) in the metric L_qp(R) for 1 < q ≤ p < ∞.     

A characterization of P2(μ) ≠ L2(μ)

It is shown that P²(μ) ≠ L²(μ) if and only if there exists a probability measure ν with ν ⊥ μ and 6pz.dfnc;_1,ν ? Cz.dfnc;pz.dfnc;_2,μ for all polynomials p and a fixed constant C < ∞. The relationship between this method and theorems of Berger and of Carey and Pincus concerning rationally cyclic vectors for powers of nonnormal subnormal Operators is examined.     

Orienting Matroids Representable Over BothGF(3) andGF(5)

For matroids representable over both GF(3) and GF(5) , we provide a method for constructing an orientation.     

A limit in probability in a W-algebra is unique

Let be the space of self-adjoint Segal measurable operators affiliated to a W^*-algebra (for x=∫λe_x(dλ), x e_x((−∞, −n)(n, ∞)) is a finite projection in for n large enough). The limit in probability is unique in . x_n → x in probability e_nProj ;e_n→1 strongly and (x_n − x) e_n → 0. The following proposition proves to be important in the investigation of . If 1 − ƒ is a finite projection in and 1 − e_n → 0 strongly, e_nProj , then strongly.     

Numerical computation of wavelet expansion: W error estimate

We consider the numerical procedure introduced in [1] in order to compute the expansion of a function f with respect to a compactly supported wavelet basis, for which we give an error estimate in the W^3,p norm. We also prove an interpolation estimate in the H³ norm.     

b~(2)空间及b~(2)空间上的等距映射

等距映射在空间结构的研究中起着很重要的作用,是泛函分析研究的有利工具.本文将介绍一类特殊的F空间,b~(2)空间,然后给出该空间单位球面间满等距映射的表现定理,进而得出b~(2)空间单位球面上满等距映射的线性延拓结论.     

Density of smooth maps in Wk, p(M, N) for a close to critical domain dimension

Assuming m − 1 < kp < m, we prove that the space C ^∞(M, N) of smooth mappings between compact Riemannian manifolds M, N (m = dim M) is dense in the Sobolev space W ^k,p (M, N) if and only if π _m−1(N) = {0}. If π _m−1(N) ≠ {0}, then every mapping in W ^k,p (M, N) can still be approximated by mappings M → N which are smooth except in finitely many points.     

Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl2n+2,0(R)*

Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domain Ω Rⁿ⁺¹ × Rⁿ⁺¹ centered at the origin with values in the Clifford algebra Cl_2n+_2,0(R) is proved. As a corollary, Almansi-type decomposition theorem for biharmonic functions of degree k is given.