Stability under composition of some classes of functions and applications to homogenization theory

In this letter we show that in contrast to what has been done so far in the deterministic homogenization theory, we can solve nonlinear homogenization problems in a general way by leaning solely on a single assumption. We also show that this assumption is suitable for finding particular solutions (such as almost periodic ones) of some partial differential equations.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.     

Characterizations of some classes of relatively regular operators

A bounded linear operator A on a Banach space is called relatively regular, if there is a bounded linear operator B such that ABA=A. In this case B is called a g₁-inverse of A. In this paper we characterize some classes of relatively regular operators A via the set {B₁-B₂:B₁ and B₂ are g₁-inverses of A}.     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

Characterizations of some polynomial variance functions byd-pseudo-orthogonality

From a notion ofd-pseudo-orthogonality for a sequence of polynomials (d ∈ {2,3, …}), this paper introduces three different characterizations of natural exponential families (NEF's) with polynomial variance functions of exact degree 2d-1. These results provide extended versions of the Meixner (1934), Shanbhag (1972), 1979) and Feinsilver (1986) characterization results of quadratic NEF's based on classical orthogonal polynomials. Some news sets of polynomials with (2d-1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions.     

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

Carleson measures and some classes of meromorphic functions

For let be the Möbius transformation defined by , and let be the Green's function of the unit disk . We construct an analytic function belonging to for all , , but not belonging to meromorphic in and for any , . This gives a clear difference as compared to the analytic case where the corresponding function spaces ( and ) are same.

     

N-widths for some classes of periodic functions with boundary conditions

For some classes of periodic functions with boundary conditions the strong asymptotics of the Kolmogorov's, the Gelfand's, and linear N-widths are obtained. The project supported by Doctoral Program Foundation of Institution of Higher Education and YNSFC.     

Extremal functions for functionals on some classes of analytic functions

It is shown that the theorem of Carathéodory and Toeplitz on the characterization of the Taylor coefficients of analytic functions with positive real part can be applied to extremal problems in several classes of analytic functions.     

Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications

We obtained new exact inequalities that estimate the L _?? -norm of the Riesz derivative D ^?? f of a function f defined on $ {\mathbb{R}^m} $ in terms of the uniform norm of the function itself and the L _s -norm of the function acted by the Laplace operator. On a class of functions f such that ||??f||s ?? 1, we solved the problem of approximation of an unbounded operator D ^?? by bounded ones and the problem of optimal recovery of the operator D ^?? on elements of this class given with known error.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

One-sided N-widths of some classes of periodic functions

In this paper,we determine the exact estimation of one-sided n-widthsd_n~+(w_q~r(p_r),L_1).q=1,∝,where W_q~r(P_r)is a class of differentiable 2π-periodic functions,and is determined by a differential opera-tor P_r(D)=D~σ∏_(j=t)~l(D-α_j),σ,j∈N,σ≥2,x_j≠0,x_j∈R.