On converse approximation theorems

We establish sufficient conditions for obtaining a strong converse inequality of type B in terms of a unified K-functional for a sequence of linear positive operators (Ln)_n?1, . This K-functional, introduced by Guo et al. (see, e.g., [S. Guo, Q. Qi, Strong converse inequalities for Baskakov operators, J. Approx. Theory 124 (2003) 219-231]), will be considered for more general weight functions. As applications we investigate the situation for Baskakov type operators and Szász-Mirakjan type operators.     

Direct and converse results for Stancu operator

In this paper we establish direct and inverse theorems for Stancu operator. Some other approximation properties of these operators are also given.     

Direct and converse imbedding theorems for seminormed spaces

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Direct and converse theorems of the theory of approximation in the metric of Lp for 0 < p < 1

A direct theorem of the Jackson type and several converse theorems are established for the approximation of periodic functions of period 2Π by trigonometrical polynomials in the metric of L_p, 0 < p < 1.     

A converse result for approximation by a weighted Szász-Mirakyan operator

We prove that the weighted error of approximation by the Szász-Mirakyan-type operator introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to previous results for Bernstein-type operators obtained by Ditzian-Ivanov and Szabados. Research supported by Hungarian Scientific Research Fund (OTKA), Grant No. T-049196.     

Direct and strong converse theorems for a general sequence of positive linear operators

We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szász–Mirakyan operators.     

Direct local and global approximation theorems for some linear positive operators

In this paper we establish direct local and global approximation theorems for Baskakov type operators and Szasz - Mirakjan type operators, respectively.     

On converse Lyapunov theorems for fluid network models

We consider the class of closed generic fluid network (GFN) models, which provides an abstract framework containing a wide variety of fluid networks. Within this framework a Lyapunov method for stability of GFN models was proposed by Ye and Chen. They proved that stability of a GFN model is equivalent to the existence of a functional on the set of paths that is decaying along paths. This result falls short of a converse Lyapunov theorem in that no state-dependent Lyapunov function is constructed. In this paper we construct state-dependent Lyapunov functions in contrast to path-wise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem. To resolve this problem we introduce the class of strict GFN models by forcing closed GFN models to satisfy a concatenation and a semicontinuity condition. For the class of strict GFN models we define a state-dependent Lyapunov function and show that a converse Lyapunov theorem holds. Finally, it is shown that common fluid network models, like general work-conserving and priority fluid network models as well as certain linear Skorokhod problems define strict GFN models.     

Direct and inverse theorems of approximation theory for a generalized modulus of smoothness

An asymmetric operator of generalized translation is introduced in this paper. Using this operator, we define a generalized modulus of smoothness and prove direct and inverse theorems of approximation theory for it.     

Direct and inverse approximation theorems in Banach spaces and Fourier transforms

An author's summary of a thesis submitted for the degree of Doctor of Physico-Mathematical Sciences. The dissertation was defended December 19, 1969, at a session of the scientific council of the Mechanical-Mathematics Faculty of the M. V. Lomonosov Moscow State University. The official opponents were Professors V. K. Dzyadyk, B. M. Levitan, and S. B. Stechkin, all Doctors of physico-mathematical sciences.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 765–778, June, 1970.     

Some common fixed point theorems for Banach operator pairs with applications in best approximation

The ordered pair (T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair if the set F(I) of fixed points of I is T-invariant i.e. T(F(I))⊆F(I). Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149-155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466-1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241-244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249-252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351], Shahzad [N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45] and of few others.     

Saturation theorems for interpolation and the Bernstein-Schnabl operator

We shall study properties of box spline operators: cardinal interpolation, convolution, and the Bernstein-Schnabl operator. We prove the saturation theorem.

     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.