Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion

Let B _?? ^p , 1 ?? p < ??, be the space of all bounded functions from L _p (?) which can be extended to entire functions of exponential type ??. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ?? B _?? ^p without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(W _p ^r (?)) are determined up to a logarithmic factor.     

Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric

We study best M-term trigonometric approximations and best orthogonal trigonometric approximations for the classes B _pθ ^r and W _pα ^r of periodic functions of several variables in the uniform metric.     

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

Extension of functions from Sobolev spaces

By definition, the domain Ω ??ⁿ belongs to the class EW _p ^l if there exists a continuous linear extension operator . An example is given of a domain Ω ??² with compact closure and Jordan boundary, having the following properties: (1) The curve ?Ω is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) Ω ε EW _p ¹ for p<2. and Ω ? EW _p ¹ for p?2. (3) for p>2 and for p?2.     

Kolmogorov widths of weighted Sobolev classes on closed intervals

In this paper, we estimate the asymptotics of the Kolmogorov widths of weighted Sobolev classes in the metric of L _p . We establish the relationship between the width of the set W _∞,g ¹ and the approximation of the antiderivative function g by piecewise constant functions.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Determinants of parabolic bundles on Riemann surfaces

LetX be a compact Riemann surface andM _s ^p (X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onM _s ^p (X). We obtain a natural metrized holomorphic line bundle onM _s ^p (X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

Description of traces of functions in the Sobolev space with a Muckenhoupt weight

We characterize the trace of the Sobolev space W _p ^l (? ⁿ , γ) with 1 < p < ∞ and weight γ ∈ A _p ^loc (? ⁿ ) on a d-dimensional plane for 1 ≤ d < n. It turns out that for a function φ to be the trace of a function f ∈ W _p ^l (? ⁿ , γ), it is necessary and sufficient that φ belongs to a new Besov space of variable smoothness, $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ , constructed in this paper. The space $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ is compared with some earlier known Besov spaces of variable smoothness.     

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ( $\mathbb{D}$ ) ? $\mathbb{D}$ . We give new characterizations of the bounded and compact weighted composition operators W _ψ,? from the Hardy spaces H ^p , 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A _α ^p , α > ? 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W _ψ,? f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H ¹ as well as of compactness for H ^p , 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.     

Approximation of classes of convolutions by linear operators of special form

A parametric family of operators G _ρ is constructed for the class of convolutions W _p,m (K) whose kernel K was generated by the moment sequence. We obtain a formula for evaluating $E(W_{p,m} (K);G_\rho )_p : = \mathop {\sup }\limits_{f \in W_{p,m} (K)} \left\| {f - G_\rho (f)} \right\|_p .$ . For the case in which W _p,m (K)=W _r,β ^p,m , we obtain an expansion in powers of the parameter ?=?ln ρ for E(W _p,m ^r,β ; G _ρ,r ) _p , where β ∈ ?, γ > 0, and m ∈ ?, while p = 1 or p = ∞.     

Reduced Load Equivalence under Subexponentiality

The stationary workload W _A+B ^φ of a queue with capacity φ loaded by two independent processes A and B is investigated. When the probability of load deviation in process A decays slower than both in B and $e^{ - \sqrt x } $ , we show that W _A+B ^φ is asymptotically equal to the reduced load queue W _A ^φ?b , where b is the mean rate of B. Given that this property does not hold when both processes have lighter than $e^{ - \sqrt x } $ deviation decay rates, our result establishes the criticality of $e^{ - \sqrt x } $ in the functional behavior of the workload distribution. Furthermore, using the same methodology, we show that under an equivalent set of conditions the results on sampling at subexponential times hold.     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.     

On the equality of Kolmogorov and relative widths of classes of differentiable functions

We obtain sharper estimates of the remainders in the expression for the least value of the multiplier M for which the Kolmogorov widths d _n (W _C ^r , C) and the relative widths K _n (W _C ^r ,MW _C ^j ,C) of the class W _C ^r with respect to the class MW _C ^j , j < r, where r ? j is odd, are equal.     

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles S _r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\) , assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I ^G and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifolds TM and S _r M.     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.