Rate of approximation for functions of bounded variation by integral operators

We estimate pointwise convergence rates of approximation for functions of bounded variation and for functions which are exponentially bounded and locally of bounded variation. The approximation is through the operation of a sequence of integral operators with not necessarily positive kernel functions. The general result is then applied to deduce estimates for particular operators, such as Beta operators, Fourier–Legendre operators, Picard operators, and Gauss–Weierstrass operators.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

A sharp estimate for extremal functions

We prove a sharp pointwise estimate for extremal functions of invariant subspaces of some weighted Bergman spaces on the unit disk. The allowed weights include standard radial weights and logarithmically subharmonic weights.

     

Rate of pointwise approximation for locally bounded functions by Szász operators

In this paper the asymptotic behavior of Szász operators for locally bounded functions f is studied at points x where f(x+) and f(x−) exist. An asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory. The estimate essentially is the best possible for continuous points of f.     

Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation

The main result says that the generator of any uniformly bounded composition operator acting between Banach algebras of functions of bounded n-th variation is an affine function with respect to the function variable.     

Degree of approximation to function of bounded variation by Bézier variant of MKZ operators

Guo (Approx. Theory Appl. 4 (1988) 9-18) introduced the integral modification of Meyer-Konig and Zeller operators and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Bézier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364-376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106-107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng.     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A uniform ergodic Theorem for certain Markov operators on Lipschitz functions on bounded metric spaces

Summary Some sufficient conditions are given for uniform convergence of the iterates of transition operators associated with random contractions of a bounded metric space.Supported by National Science Foundation grant GP-7335.     

Rate of approximation for functions with derivatives of bounded variation

We estimate pointwise convergence rates of approximation for functions with derivatives of bounded variation and for functions which are exponentially bounded and have derivatives locally of bounded variation. The approximation is made through the operation of a sequence of integral operators with not necessarily positive kernel functions. The general result is then applied to deduce estimates for Beta operators, Hermite-Fejér operators, Picard operators, Gauss-Weierstrass operators, Baskakov operators, Mirakjan-Szász operators, Bleimann-Butzer-Hahn operators, Phillips operators, and Post-Widder operators.     

A NOTE ON MEYER—KONIG AND ZHLLER OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

In the present note we give the correct and improved estimate on the rate of convergence of integrated Meyer-Konig and Zeller operators for function of bounded variation.     

A sharp function estimate for multilinear singular integral operators

A variant sharp function estimate is established for the multilinear singular integral operators. Two applications of this estimate are also given. Supported by the NSF of China (19701039)     

A sharp function estimate for multilinear singular integral operators

A variant sharp function estimate is established for the multilinear singular integral operators. Two applications of this estimate are also given.     

Locally Lipschitz composition operators in spaces of functions of bounded variation

We give a necessary and sufficient condition on a function \({f:\mathbb{R}\to\mathbb{R}}\) under which the composition operator (Nemytskij operator) F defined by \({Fh=f\circ h}\) acts in the spaces \({BV_\varphi[a,b], HBV[a,b], {\rm and} \,RV_\varphi[a,b]}\) and satisfies a local Lipschitz condition. While the proof of sufficiency consists in a straightforward calculation, the proof of necessity builds on nontrivial arguments like Helly’s selection principle or the Arzelà–Ascoli compactness criterion.     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.