A characterization and equations for minimal shape-preserving projections

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by the space of all bounded linear operators from X into V; let be the set of all projections in . For a given , we denote by the set of operators such that PSS. When , we characterize those for which P is minimal. This characterization is then utilized in several applications and examples.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

Orthonormal polynomials with exponential-type weights

Let and let w_ρ(x)|x|^ρexp(-Q(x)), where and is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight , obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate A_n(x) and B_n(x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight .     

Weyl sums in with digital restrictions

Let be a finite field and consider the polynomial ring . Let . A function , where G is a group, is called strongly Q-additive, if f(AQ+B)=f(A)+f(B) holds for all polynomials with degB<degQ. We estimate Weyl sums in restricted by Q-additive functions. In particular, for a certain character E we study sums of the form

where is a polynomial with coefficients contained in the field of formal Laurent series over and the range of P is restricted by conditions on f_i(P), where f_i (1ir) are Q_i-additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form

with . Sums of this shape are treated by applying the kth iterate of the Weyl–van der Corput inequality and studying higher correlations of the functions f_i. With these Weyl sum estimates we show uniform distribution results.     

n-Star modules over ring extensions

We give conditions under which an n-star module extends to an n-star module, or an n-tilting module, over a ring extension R of A. In case that R is a split extension of A by Q, we obtain that is a 1-tilting module (respectively, a 1-star module) if and only if is a 1-tilting module (respectively, a 1-star module) and generates both and (respectively, generates ), where is an injective cogenerator in the category of all left A-modules. These extend results in [I. Assem, N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 (1998) 1547–1555; K.R. Fuller, *-Modules over ring extensions, Comm. Algebra 25 (1997) 2839–2860] by removing the restrictions on R and Q.     

Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians

In this paper, we prove a combinatorial rule describing the restriction of any irreducible representation of U(n+m) to the subgroup U(n)×U(m). We also derive similar rules for the reductions from SU(n+m) to S(U(n)×U(m)), and from SU(n+m) to SU(n)×SU(m). As applications of these representation-theoretic results, we compute the spectra of the Bochner–Laplacian on powers of the determinant bundle over the complex Grassmannian . The spectrum of the Dirac operator acting on the spin Grassmannian is also partially computed. A further application is given by the determination of the spectrum of the Hodge–Laplacian acting on the space of smooth functions on the unit determinant bundle over .     

Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form , where , states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in , or which forms a frame for , is that the density condition is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set which is -shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L²(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set with the property that the Gabor system associated with the same parameters a,b and the window g=χ_E, forms a tight frame for L²(S).     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over [a,b], b_ijk are real numbers, and the points t_k belong to [a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).     

Gradient estimates and Jackson’s theorem in spaces related to measures

Jackson’s theorem is established in a new kind of holomorphic function space Q_μ related to measures in any starlike circular domain in . Particularly, the result covers many spaces including BMOA, Q_p, Q_K, and F(p,q,s) spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem.     

Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph as follows: Take G L(G) as vertices of and join two distinct vertices x and y whenever [x,_ky]≠1 and [y,_kx]≠1 for all positive integers k. We call , the Engel graph of G. In this paper we study the graph theoretical properties of .     

Recursively enumerable sets of polynomials over a finite field

We prove that a relation over is recursively enumerable if and only if it is Diophantine over . We do this by first constructing a model of in , where n is represented by Zⁿ. In a second step, we show that it suffices to eliminate a bounded universal quantifier. Then finally, the hardest part of the proof is to show that we can eliminate this quantifier.     

The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra of dimension attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra and a primitive root of unity q of order n² is equivalent to Lusztig's small quantum group (under some conditions on n). We also give a conceptual construction of using the notion of de-equivariantization of tensor categories.     

On an extremal problem of Selberg

We consider the minimum value m_p of the product under the conditions that and . This problem stems from estimation of the Selberg integral appeared in the study of integral-valued entire functions [A. Selberg, Über einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941) 159–170]. We give upper and lower estimates of m_p : . This, in particular, slightly improves the earlier lower estimate given by Selberg: logm_p>p·0.31654924….     

Orthogonal polynomials for exponential weights on , II

Let I=[0,d), where d is finite or infinite. Let W_ρ(x)=x^ρexp(-Q(x)), where and Q is continuous and increasing on I, with limit ∞ at d. We obtain further bounds on the orthonormal polynomials associated with the weight , finer spacing on their zeros, and estimates of their associated fundamental polynomials of Lagrange interpolation. In addition, we obtain weighted Markov and Bernstein inequalities.     

Zeros of Bernoulli-type functions and best approximations

The zero sets of (D+a)ⁿg(t) with in the (t,a)-plane are investigated for and .The results are used to determine entire interpolations to functions , which give representations for the best approximation and best one-sided approximation from the class of functions of exponential type η>0 to .     

Interpolation and approximation in

Assume a standard Brownian motion W=(W_t)_t[0,1], a Borel function such that f(W₁)L₂, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a_X(f(X₁);τ) can be used to describe the L₂-error in discrete time simulations of the martingale generated by f(W₁) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.