On Anastassiou's Generalizations of the Ostrowski Inequality and Related Results

We provide a generalization of a recent result of Anastassiou related to thewell-known Ostrowski inequality, as well as some related results. Ourresults subsume, extend, and consolidate a number of known results.     

On a discrete Korovkin theorem

In [G. A. Anastassiou, A discrete Korovkin theorem, J. Approx. Theory 45 (1985), pp. 383–388, Theorem 3], a discrete Korovkin theorem was given. We restate the theorem here and its proof, correcting a mistake in the above reference.     

Some Shift-Invariant Integral Operators, Univariate Case, Revisited

In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and ⁿ , n 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over ⁿ that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties.     

Shape-preserving bivariate polynomial approximation inC([−1,1]×[−1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω2-modulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

Higher order Ostrowski type inequalities over Euclidean domains

The classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to functions on general domains using the L^∞ norm of its nth partial derivatives. For radial functions on balls the inequality is sharp.     

Cosine families generated by second-order differential operators on W1,1(0, 1) with generalized Wentzell boundary conditions

Using the abstract framework [Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations with generalized Wentzell boundary conditions. Journal of Differential Equations, 207, 1–20.] we show that certain second-order differential operators with generalized Wentzell boundary conditions generate cosine families and hence also analytic semigroups on W^1,1(0,1). This complements the main result [Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht, E. and Romanelli, S., 2003, General Wentzell boundary conditions and analytic semigroups on W^{1, p} (0,1). Applicable Analysis, 82, 927–935.] on the generation of an analytic semigroup by the second derivative with generalized Wentzell boundary conditions on W^{1, p} (0,?1) for 1<p<∞.     

General weighted two-point Radau and Gauss and three-point Lobatto quadrature formulae for functions in L p spaces

We use a recently obtained generalization of the weighted Montgomery identity to obtain new sharp weighted generalizations of Ostrowski type in- equalities, as well as weighted 2-point Radau and Gauss and 3-point Lobatto quadrature formulae for functions whose first derivative belong to the L _p spaces.     

The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

Solution générale de l'équation fonctionnellef[x +yf(x)]=f(x)f(y)

Sans résuméDédié à A. M. Ostrowski à l'occasion de son 75^e anniversaire     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

An extension of a uniqueness theorem of J. A. Goldstein

A general uniqueness theorem for the abstract Cauchy problem x′(t) = f(t, x(t)); x(t) = P(t) + o(μ(t)) as t ↓ 0 is proved extending a result due to Goldstein. Applications of this theorem to a partial differential equation of a generalized Euler-Poisson-Darboux type are also discussed.     

Second-order subdifferentials of C 1,1 functions and optimality conditions

We present second-order subdifferentials of Clarke's type of C ^1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ? ⁿ , considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C ^1,1 data are obtained.     

On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications

An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫_a^b ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.     

Shape-preserving bivariate polynomial approximation inC([?1,1]×[?1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω ₂ ^ρ of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function. The result extends that in univariate case-of D. Leviatan in [5–6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

Some angularity and inertia theorems related to normal matrices

Motivated by the definition of the inertia, introduced by Ostrowski and Schneider, a notion of angularity of a matrix is defined. The angularity characterizes the distribution of arguments of eigenvalues of a matrix. It is proved that if B and C are nonsingular matrices, then B¹AB and C¹AC have the same angularity provided they are diagonal. Some well-known inertia theorems (e.g. Sylvester's law) have been deduced as corollaries of this result. The case when C is permitted to be singular is discussed next. Finally, we prove that (a) any linear transformation T, on the set of n by n complex matrices, mapping Hermitian matrices into themselves and preserving the inertia of each Hermitian matrix is of the form T(A)=C¹AC or T(A)=C^1LA′C where C is some nonsingular matrix, and (b) any linear transformation T mapping normal matrices into normal matrices and preserving the angularity of each normal matrix is also of one of the above forms, but with C=kU where k≠0 and U is unitary.     

Identities for the Hankel transform and their applications

In the present paper the authors show that iterations of the Hankel transform with Kν-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.     

Unitary similarity of projectors

Summary We deal with linear operators acting in a finite dimensional complex Hilbert space. We show that there exists a simple canonical form for projectors (not necessarily orthogonal) under unitary similarity. As a consequence we obtain a simple test for unitary similarity of projectors. IfP is a projector we show thatP andP ^* are unitarily similar. We also determine the isomorphism type of the algebra generated by the projectorsP andP ^*.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

M-matrices leading to semiconvergent splittings

An M-matrix as defined by Ostrowski is a matrix that can be split into A = sI ? B, s > 0, B ? 0 with s ? ρ(B), the spectral radius of B. M-matrices with the property that the powers of T = (1/s)B converge for some s are studied and are characterized here in terms of the nonnegativity of the group generalized inverse of A on the range space of A, extending the well-known property that A^{? 1} ? 0 whenever A is nonsingular.     

Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense

New inequalities of Ostrowski type for functions whose derivatives in absolute value are $s$-convex in the second sense are obtained.