Generalization of Ostrowski and Čebyšev type inequalities involving many functions

Generalized Ostrowski and ?eby?ev type inequalities involving many functions on time scales are derived that generalize some existing and classical inequalities with some applications for generalized polynomials.     

Monotone and Čebyšev arcs in hyperspaces

A set in a metric space is called a ?eby?ev set if it contains a unique “nearest neighbour” to each point of the space. In this paper we introduce the concept of a monotone arc of convex sets and show that compact monotone arcs have the ?eby?ev property in the hyperspace of compact strictly convex sets. In the hyperspace of compact convex sets only certain monotone arcs are ?eby?ev ; these are characterized. Results are also obtained for affine segments and for noncompact monotone arcs.     

Some trace inequalities of Čebyšev type for functions of operators in hilbert spaces

Some trace operator inequalities for synchronous functions that are related to the ?eby?ev inequality for sequences of real numbers are given.     

Čebyšev-Grüss-type inequalities revisited

We generalize and improve several inequalities of the ?eby?ev-Grüss-type using least concave majorants of the moduli of continuity of the functions involved. Our focus is on normalized positive linear functionals. We discuss a problem posed by the two Gavreas and also give the solution of a stronger one. In a section about the non-multiplicativity of positive linear operators it is demonstrated that the previous use of second moments is not quite the right choice. This is documented in the case of the classical Hermite-Fejér and de La Vallée Poussin convolution operators.     

On the distribution of points of maximal deviation in complex Čebyšev approximation

f — , . p _n (f) f . , n+2 , f–p _n (f) . , n . , .

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On the distribution of points of maximal deviation in complex ebyev approximation

     

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.     

A closed formula for the čebyšev barycentric weights of optimal approximation in H2

The barycentric formula has several advantages over other means of evaluating the polynomial interpolating a function betweenn points in an interval. In particular, it is much more stable for sets of points clustered at the extremities of the interval, as are all the sets guaranteeing a good approximation forn sufficiently large. Also, it requires onlyO(n) operations for every function to be interpolated, once some weights, which depend only on the points, have been computed. Computing those weights usually requiresO(n²) operations; for ebyev points, however,O(n) operations suffice. We show here that all the above is also true for the optimal evaluation of functionals in H² by giving a closed formula for the corresponding weights.     

Functional inequalities and subordination: stability of Nash and Poincaré inequalities

We show that certain functional inequalities, e.g. Nash-type and Poincaré-type inequalities, for infinitesimal generators of C ₀ semigroups are preserved under subordination in the sense of Bochner. Our result improves earlier results by Bendikov and Maheux (Trans Am Math Soc 359:3085–3097, 2007, Theorem 1.3) for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.     

The Bernstein–Orlicz norm and deviation inequalities

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

The Hasegawa–Petz mean: Properties and inequalities

We study a family of means introduced by H. Hasegawa and D. Petz (1996) [13]. Properties with respect to the parameter, such as monotonicity and logarithmic concavity, further, monotonicity and concavity in the mean variables are shown. Besides, the comparison between the Hasegawa–Petz mean and the geometric mean is completely solved. The connection to earlier results on operator monotonicity and some applications are also discussed.     

Intersections of arithmetic Hirzebruch–Zagier cycles

We establish a close connection between the intersection multiplicities of three arithmetic Hirzebruch–Zagier cycles and the Fourier coefficients of the derivative of a certain Siegel–Eisenstein series at its center of symmetry. Our main result proves a conjecture of Kudla and Rapoport.     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

Interpolation and Φ-moment inequalities of noncommutative martingales

This paper is devoted to the study of Φ-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative Φ-moment analogues of martingale transformations, Stein’s inequalities, Khintchine’s inequalities for Rademacher’s random variables, and Burkholder–Gundy’s inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.     

Gagliardo–Nirenberg inequalities and non-inequalities: The full story

We investigate the validity of the Gagliardo–Nirenberg type inequality

(1)${6f6}_{W^{s,p}(\mathrm{\Omega })}?{6f6}_{W^{s_1,p_1}(\mathrm{\Omega })}^{\theta }{6f6}_{W^{s_2,p_2}(\mathrm{\Omega })}^{1?\theta },$

with $\mathrm{\Omega }?{\mathbb{R}}^N$. Here, $0\le s_1\le s\le s_2$ are non negative numbers (not necessarily integers), $1\le p_1,p,p_2\le \infty$, and we assume the standard relations

$s=\theta s_1+(1?\theta )s_2,1/p=\theta /p_1+(1?\theta )/p_2\text{ for some }\theta \in (0,1).$

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when $s_1,s_2,s$ are integers. It turns out that (1) holds for “most” of values of $s_1,\dots ,p_2$, but not for all of them. We present an explicit condition on $s_1,s_2,p_1,p_2$ which allows to decide whether (1) holds or fails.