We extend the interval and fuzzy gH-differentiability to consider interval and fuzzy valued functions of several variables and to include directional gH-differentiability; the proposed setting is ...more general than the existing definitions in the literature and allows a unified view of total and direction gH-differentiability and for the computation of partial gH-derivatives, directional gH-derivative and level wise gH-differentiability in the fuzzy valued case. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well known tangency between a differentiable function and its tangent plane. The proposed new setting allows an analysis of conditions for local optimality (dominance with respect to interval and level-wise partial orders well known in the literature) in terms of directional gH-derivatives, including concepts of local convexity, and to formulate KKT-like conditions for non-dominated solutions in constrained optimization problems.
The classical Szegö–Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of ...recurrence coefficients for the orthogonal polynomials determined by the measure. The present paper constructs orthogonal polynomials on the torus of arbitrary finite dimension in order to prove theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings. The results are applied to the one-dimensional Schrödinger equation in impedance form to yield a new trace formula valid for piecewise constant impedance, a case where the classical trace formula breaks down. As a byproduct, the analysis gives an explicit formula for the Taylor coefficients of a bounded holomorphic function on the open disk in terms of its continued fraction expansion.
We introduce a pseudometric TV on the set MX of all functions mapping a rectangle X on the plane R2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and ...{gj} of functions from MX are such that {fj} is pointwise precompact on X, {gj} is pointwise convergent on X with the limit g∈MX, and the limit superior of TV(fj,gj) as j→∞ is finite, then a subsequence of {fj} converges pointwise on X to a function f∈MX such that TV(f,g) is finite. One more pointwise selection theorem is given in terms of total ε-variations (ε>0), which are approximations of the total variation as ε→0.
We give a general construction of entire functions in d complex variables that vanish on a lattice of the form
for an invertible complex-valued matrix. As an application, we exhibit a class of ...lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in
. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.
A special class of approximations of continuous functions of several variables on the unit coordinate cube is investigated. The class is constructed using Kolmogorov’s theorem stating that functions ...of the indicated type can be represented as a finite superposition of continuous single-variable functions and another result on the approximation of such functions by linear combinations of quadratic exponentials (also known as Gaussian functions). The effectiveness of such a representation is based on the author’s previously proved assertion that the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions. It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.
Solutions of the Dirichlet and Neumann problems for multidimensional singular elliptic equations in an infinite domain can be found in explicit forms in recent works of the authors. In this paper, a ...problem with mixed conditions, which is a natural generalization of the previously considered Dirichlet and Neumann problems, is studied. To prove the existence of a unique solution to the problem, we use the representation of the multiple Lauricella hypergeometric function at limiting values of the variables and a new representation for multiple improper integrals which generalizes the well-known representation from the handbook of Gradshtein and Ryzhik.
In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, ...namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.
The authors propose an algorithm to construct Chebyshev approximation for functions of several variables by a generalized polynomial as a limiting approximation in the norm of space L
p
as p → ∞. It ...is based on serial construction of power-average approximations using the least squares method with variable weight function. The convergence of the method provides an original way to consistently refine the values of the weight function, which takes into account the results of approximation at all previous iterations. The authors describe the methods of calculating the Chebyshev approximation with absolute and relative errors. The results of test examples confirm the efficiency of using the method to obtain Chebyshev approximation of tabular continuous functions of one, two, and three variables.
We prove a joint generalization of Whitneyʼs C1 extension theorem and Aversa–Laczkovich–Preissʼ extension theorem. It can be roughly described as a theorem on extendibility of a differentiable ...function defined on a closed subset of Rn to a differentiable function on Rn with preserving the continuity of the derivative. Other results on C1 and differentiable extensions from “thick” subsets of Rn are also proved.
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