For set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$, we study approximation based on the metric approach that includes metric linear combinations, metric selections and weighted metric integrals. In our earlier papers we considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity $x$ of the set-valued function $F$ is $F(x)$, the limit set at a jump point was earlier described in terms of the metric selections of the multifunction. Here we show that, under certain assumptions on $F$, the limit set at $x$ equals the metric average of the left and the right limits of $F$ at $x$, thus extending the case of real-valued functions.

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive definite functions f on G such that f(0) = 1, the positive part of f is supported in X, and its negative part is supported in Y. In the case when X=Y, the problem is exactly the so-called Turan problem for the set X. When Y= G, i.e., there is a restriction only on the set of positivity of f, we obtain the Delsarte problem. The Delsarte problem is the sharpest Fourier analytic tool to study packing density by translates of a given master copy set, which was studied first in connection with packing densities of Euclidean balls. We give an upper estimate of the constant C(X,Y) in the situation when the set X satisfies a certain packing type condition. This estimate is given in terms of the asymptotic uniform upper density of sets in locally compact Abelian groups.

We consider approximation of functions of several variables by continuous linear splines interpolating the given function in the knots of a rectilinear lattice. For function classes defined in terms of a modulus of continuity, we give an exact es-timate for the error of approximation. In the particular case when the modulus of continuity is concave and the distance between points in Rd is measured in the l(p)-norm with 1 <= P <= 3, we calculate an explicit value of the exact approximation error on the class. Surprisingly, the behavior changes dramatically if P > 3. We show that the our estimate is no longer true, in general, when P > 3. We also consider approximation of a first derivative of a function by the corre-sponding derivative of the linear continuous spline and obtain an upper estimate for the error of approximation for an arbitrary modulus of continuity, all 1 <= P <= infinity, and triangulations of the staircase type.

In this paper we consider a generalization of the Bernstein–Durrmeyer operator where the integrals are taken with respect to measures that may vary from term to term. This construction is more general than the one considered by the first named author and her coauthors earlier, and it includes a number of well-known operators of Bernstein type as particular cases. We give conditions on the collections of measures that guarantee pointwise convergence at a point of continuity of a function.

Let \(A\) be the infinitesimal generator of a strongly continuous contraction semigroup in a Hilbert space \(H\). We give an upper estimate for the best approximation of the operator \(A\) by bounded linear operators with a prescribed norm in the space \(H\) on the class \(Q_2 = \{x\in \mathcal{D}(A^2) : \|A^2 x\| \leq 1\}\), where \(\mathcal D(A^2)\) denotes the domain of \(A^2\).

In this paper, we introduce Sz´asz-Mirakjan-Durrmeyer operators and BaskakovDurrmeyer operators with respect to an arbitrary measure, thus extending this natural generalization from the already studied case of the Bernstein-Durrmeyer operators to these two closely related cases. We establish convergence of the new operators, namely, pointwise convergence at each point of continuity of a function in the support of the measure, and uniform convergence in every compact set in the interior of the support of the measure. In case when the measure is finite, we also show convergence in the corresponding weighted Lp-spaces, 1 ≤ p < ∞. In the latter case, we also give estimates for the rate of convergence in terms of a K-functional.

In this paper, we give an overview of operators of Durrmeyer type with respect to arbitrary measure. Our construction includes the Bernstein–Durrmeyer operator, the Szász–Mirakjan–Durrmeyer operator, and the Baskakov–Durrmeyer operator with respect to arbitrary measure. We are particularly interested in the convergence of the operators. We discuss the uniform and the pointwise convergence as well as convergence in the corresponding weighted \(L^p\)-spaces. A new result is the statement on the \(L^p\)-convergence of the Szász–Mirakjan–Durrmeyer operator and the Baskakov–Durrmeyer operator without additional restrictions on the measure.

Turán’s problem for ℓ-1 radial, positive definite functions is to determine the maximal possible value of the integral $$\int_{\mathbb{R}^{d}}f(\mathbf{x})\,d\mathbf{x}$$ over the class of continuous, positive definite, ℓ-1 radial functions $$f$$ on $$\mathbb{R}^{d}$$ with support in an ℓ 1-ball and with f(0) = 1. This problem can be reformulated as a Turán problem for functions defined on [0, ∞). The reformulation uses the pair of transformations $$(h_{d},m_{d})$$ on (0, ∞) that appears in studies on Fourier transforms of ℓ-1 radial functions.We consider a discrete Turán problem for the pair $$(h_{d},m_{d})$$ . We solve this problem for dimensions 2, 3, 5, and 7 and formulate a conjecture for other dimensions. Our considerations lead to an investigation of certain monotonicity properties of spherical Bessel functions.

The Bernstein–Durrmeyer operator with respect to arbitrary measure is a modification of the classical Bernstein operator for functions from the corresponding weighted Lq-spaces on a simplex in Rd. As a first step in studying convergence of this operator, we consider uniform convergence. We prove that uniform convergence holds for all continuous functions if and only if the measure is strictly positive on the simplex. As a consequence, strict positivity of the measure is sufficient for convergence in the weighted Lq-spaces.

We establish complete asymptotic expansions in terms of certain differential operators for the Bernstein–Durrmeyer operators with Jacobi weights on the d-dimensional simplices, and for their so-called natural quasi-interpolants. Our method extensively uses spectral properties of the involved operators. In particular, we prove new identities for the eigenvalues of the operators. We also show that the obtained asymptotic expansions can be differentiated term-by-term.

In this paper we introduce a class of Bernstein–Durrmeyer operators with respect to an arbitrary measure ρ on the d-dimensional simplex, and a class of more general polynomial integral operators with a kernel function involving the Bernstein basis polynomials. These operators generalize the well-known Bernstein–Durrmeyer operators with respect to Jacobi weights. We investigate properties of the new operators. In particular, we study the associated reproducing kernel Hilbert space and show that the Bernstein basis functions are orthogonal in the corresponding inner product. We discuss spectral properties of the operators. We make first steps in understanding convergence of the operators.

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.

Turán’s problem is to determine the greatest possible value of the integral ∫ℝ df(x)dx/ f (0) for positive definite functions f (x), x ∈ ℝd, supported in a given convex centrally symmetric body D ⊂ ℝd. In this note we consider the 2-dimensional Turán problem for positive definite functions of the form f(x) = φ (∥x∥1), x ∈ ℝ2, with φ supported in [0,π].

To a function \(f \in L_2 [ - \pi ,\pi ]\) and a compact set \(Q \subset [ - \pi ,\pi ]\) we assign the supremum \(\omega (f,Q) = \sup _{t \in Q} ||f( \cdot + t) - f( \cdot )||_{L_2 [ - \pi ,\pi ]} \), which is an analog of the modulus of continuity. We denote by \(K(n,Q)\) the least constant in Jackson's inequality between the best approximation of the function f by trigonometric polynomials of degree \(n - 1\) in the space \(L_2 [ - \pi ,\pi ]\) and the modulus of continuity \(\omega (f,Q)\). It follows from results due to Chernykh that \(K(n,Q) \geqslant 1/\sqrt 2 \) and \(K(n,[0,\pi /\pi ]) = 1/\sqrt 2 \). On the strength of a result of Yudin, we show that if the measure of the set Q is less than \(\pi /n\), then \(K(n,Q) >1/\sqrt 2 \).

A new class of differential operators on the simplex is introduced, which define weighted Sobolev norms and whose eigenfunctions are orthogonal polynomials with respect to Jacobi weights. These operators appear naturally in the study of quasi-interpolants which are intermediate between Bernstein–Durrmeyer operators and orthogonal projections on polynomial subspaces. The quasi-interpolants satisfy a Voronovskaja-type identity and a Jackson–Favard-type error estimate. These and further properties follow from a spectral analysis of the differential operators. The results are based on a pointwise orthogonality relation of Bernstein polynomials that was recently discovered by the authors.

The Turán Problem for a Class of Polytopes Elena Berdysheva Mathematisches Institut, Universität Erlangen–Nürnberg berdyshe@mi.uni-erlangen.de Consider the class of positive–definite functions f on Rd with f(0) = 1 and supported in a given convex centrally symmetric body D. The Turán problem is to find the least upper bound for integrals of such functions. We solve the Turán problem in the case where D is a convex centrally symmetric polytope in Rd with the property that its translations over a lattice form a tiling of Rd. Joint work with V.V. Arestov (Yekaterinburg, Russia). About Generalised Energy Functionals, Equidistributed Point Sets and Invariance Principles Johann S. Brauchart Institut für Mathematik A, Technische Universität Graz brauchart@finanz.math.tu-graz.ac.at Heuristics expects that point sets with extremal energy are good point sets for numerical integration with Chebyscheff–type quadrature formula. A general class of energy functionals with some of their properties will be introduced. A generalisation of Stolarsky’s invariance principle is used to obtain invariance principles for this class of energy functionals.