Group invariance and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.     

Interpolation of nonlinear operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We show that Peetre’s classical interpolation theorem in weighted L _p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.     

The rate of pointwise approximation of positive linear operators based on <Emphasis Type="Italic">q</Emphasis>-integer

The paper deals with positive linear operators based on q-integer. The rate of convergence of these operators is established. For these operators, we present Voronovskaya-type theorems and apply them to q-Bernstein polynomials and q-Stancu operators.     

(p,q)‐Generalization of Szász–Mirakyan operators

In this paper, we introduce new modifications of Szász–Mirakyan operators based on (p,q)‐integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p = 1, the previous results for q‐Sz ász–Mirakyan operators are captured. Copyright © 2015 John Wiley & Sons, Ltd.     

Phase‐field systems for multi‐dimensional Prandtl–Ishlinskii operators with non‐polyhedral characteristics

Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase‐field models in which hysteresis non‐linearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so‐called Prandtl–Ishlinskii operators. For these operators, the corresponding phase‐field systems are in the multi‐dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi‐dimensional Prandtl–Ishlinskii operators having non‐polyhedral convex characteristicsets. Copyright © 2002 John Wiley & Sons, Ltd.     

The moments for <Emphasis Type="Italic">q</Emphasis>-Bernstein operators in the case 0 < <Emphasis Type="Italic">q</Emphasis> < 1

In this note we give the estimates of the central moments for q-Bernstein operators (0 < q < 1) which can be used for studying the approximation properties of the operators.     

Higher order limit q‐Bernstein operators

In this paper we give the estimates of the central moments for the limit q‐Bernstein operators. We introduce the higher order generalization of the limit q‐Bernstein operators and using the moment estimations study the approximation properties of these newly defined operators. It is shown that the higher order limit q‐Bernstein operators faster than the q‐Bernstein operators for the smooth functions defined on [0, 1]. Copyright © 2011 John Wiley & Sons, Ltd.     

Solvability criterion and representation of solutions of <Emphasis Type="Italic">n</Emphasis>-normal and <Emphasis Type="Italic">d</Emphasis>-normal linear operator equations in a Banach space

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.     

Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.     

On discrete <Emphasis Type="Italic">q</Emphasis>-beta operators

The aim of the present paper is to introduce and study the q-analogue of discrete beta operators. First, we show some approximation properties of these operators. Then, we establish some global direct error estimates for the above operators using the second order Ditzian–Totik modulus of smoothness. Finally, we define and study the limit discrete q-beta operator.     

Pseudodifferential <Emphasis Type="Italic">p</Emphasis>-adic vector fields and pseudodifferentiation of a composite <Emphasis Type="Italic">p</Emphasis>-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O _p ) of balls in O _p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O _p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.     

Modal operators on bounded commutative residuated <Emphasis Type="Italic">ℓ</Emphasis>-monoids

Bounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVá,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras. The first author was supported by the Council of Czech Government, MSM 6198959214.     

Generalized <Emphasis Type="Italic">q</Emphasis>-Baskakov operators

In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.     

Iterative algorithm for a system of variational inclusions involving <Emphasis Type="Italic">H</Emphasis>-accretive operators in Banach spaces

Summary We introduce and study a system of variational inclusions involving H-accretive operators in Banach spaces. By using the resolvent operator technique associated with an H-accretive operator, we prove the existence and uniqueness of solution for the system of variational inclusions involving H-accretive operators and construct a new iterative algorithm to approximate the unique solution.     

Korovkin type approximation theorem for BBH type operators via <Emphasis Type="Italic">J</Emphasis>-convergence

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.     

Calderón–Zygmund operators with operator‐valued kernel on homogeneous Besov spaces

We consider generalized Calderón–Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the T (1) theorem of David and Journé we prove boundedness results for such operators on vector‐valued Besov spaces. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable <Emphasis Type="Italic">sl</Emphasis>(<Emphasis Type="Italic">N</Emphasis>, ℂ) tops as Calogero-Moser systems

We show a relation between systems of integrable tops on the algebras sl(N, ℂ) and Calogero-Moser systems of N particles. We construct classical Lax operators corresponding to these systems. We show that these operators are related to certain new trigonometric and rational solutions of the Yang-Baxter equations for the algebras sl(N, ℂ) and give explicit formulas for N = 2, 3. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 355–369, March, 2009.     

Multivalued fractals in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.