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We consider Chebyshev polynomials, T n ( z ) , for infinite, compact sets e ⊂ R (that is, the monic polynomials minimizing the sup -norm, | | T n | | e , on e ). We resolve a 45 + year old conjecture of Widom that for finite gap subsets of R , his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form | | T n | | e ≤ Q C ( e ) n (where C ( e ) is the logarithmic capacity of e ) for a class of e ’s with an infinite number of components, explicitly for those e ⊂ R that obey a Parreau–Widom condition.

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an L ∞ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of R ¯ . We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widom and DCT conditions.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool.

Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint.

We obtain two-sided sharp inequalities for the uniform approximation of bivariate functions by sums of univariate functions on step polygons of the xy plane.

Two canonical polynomials generate all cubics, via linear transformations of the polynomial map and the parameter: the cubic power function, with coincident critical points, and the third Chebyshev polynomial of the first kind, with two distinct critical points. Computing the roots of any cubic boils down to inverting these fundamental maps. In the more general case of distinct critical points, we show that the roots admit a startlingly simple expression in terms of a Joukowsky map and its inverse. Marden's theorem comes as a straightforward consequence, because the roots are the images, under a Joukowsky map, of the vertices of an equilateral triangle.

In this paper, we consider a finite family of sets (geometric constraints) F1,F2,…,Fr in the Euclidean space Rn. We show under mild conditions on the geometric constraints that the “perturbation property” of the constrained best approximation from a nonempty closed set K∩F is characterized by the “convex conical hull intersection property” (CCHIP in short) at a reference feasible point in F. In this case, F is the intersection of the geometric constraints F1,F2,…,Fr, and K is a nonempty closed convex set in Rn such that K∩F≠∅. We do this by first proving a dual cone characterization of the contingent cone of the set F. Finally, we obtain the “Lagrange multiplier characterizations” of the constrained best approximation. Several examples are given to illustrate and clarify our results.

The initial value problem of stiff functional differential equations often appears in many fields such as automatic control, economics and its theoretical and algorithmic research is of unquestionable importance. The paper proposes a rigid functional equation based on the integral process method of the financial accounting measurement model of numerical analysis. This method provides a unified theoretical basis for the stability analysis of the solution of the functional differential equation encountered in the integrodifferential equation and the financial accounting fair value measurement model of investment real estate.

The hyperplane theory plays a very important role in the research of optimization, it can help us better understand and solve various optimization problems. Therefore, the development of hyperplane theory has always been concerned by the scholars. This paper mainly studies the quasi-support hyperplane, which is the generalization of a support hyperplane, on the asymmetric normed space. First, the properties of the distance from a point to a non-empty set are studied more comprehensively, the isometric property of the distance from a point to a half space and to the corresponding hyperplane is proved. Second, the problem of supremum (infimum, resp.) of a linear function on a set is reduced to the problem of upper quasi-support (lower quasi-supports, resp.) hyperplane of the set. Additionally, some equivalent conditions for the quasi-support hyperplane are given by using the theory of dual space of asymmetric normed spaces. The obtained results enrich the functional analysis in asymmetric normed spaces and have great significance to applications in optimization theory, approximation problems, complexity analysis and the other fields.