Extremalis problémák többváltozós és súlyozott polinomokra = Extremal problems for multivariate and weighted polynomials

Jól ismert hogy a többváltozós polinomok sűrűek a d-dimenziós kompakt halmazokon folytonos függvények terében. A többváltozós polinomok egy fontos részhalmaza a homogén polinomok osztálya. Igy természetesen felmerül az a kérdés, hogy igaz-e a sűrüség a homogén polinomokra? Egy ismert sejtés szerint a konvex felületeken folytonos függvények megközelíthetőek két homogén polinom összegével. A pályázat keretében két fontos új eredmény született 1) igazoltuk a sejtést tetszőleges sima ( egyértelmü támasz sikkal rendelkező) konvex testeken egyenletes normában 2) igazoltuk a sejtést teljes általánosságban Lp normában Ezen kivül általánosított Freud súlyokra vonatkozó polinom-approximációs problémákat vizsgáltunk. Itt az általánosítás azt jelenti, hogy az eredeti Freud súlyokat megszorozzuk olyan un. általánosított polinomokkal, amelyeknek csak valós gyökeik vannak. A klasszikus polinom-egyenlotlenségek analogonjait, valamint direkt és fordított approximációs tételeket bizonyítottunk. Hibabecsléseket adtunk függvények súlyozott approximációjára Freud súlyok esetén, olyan egész függvényekkel történo approximáció esetén, amelyek véges, ill. végtelen sok pontban interpolálják a függvényt. Ezek a hibabecslések olyan súlyozott folytonossági modulusokat tartalmaznak, amelyeknél a polinom-suruség nem mindig garantált | It is well known that multivariate polynomials are dense in the space of continuous functions on compact subsets of the d-dimensional space. An important family of multivariate polynomials is the space of all homogeneous polynomials. Thus it is natural to ask if the density holds for homogeneous polynomials. It has been conjectured that any function continuous on a convex surface can be approximated by sums of two homogeneous polynomials. In the framework of the present project the above conjecture was verified in two new important cases: 1) the conjecture was verified for uniform norm on arbitrary regular convex bodies, i.e., in case when the body possesses a unique tangent plane at each point of its boundary 2) the conjecture was verified in full generality in the Lp norm We also considered polynomial approximation problems on the real line with generalized Freud weights. The generalization means multiplying these weights by so-called generalized polynomials which have real roots only. Analogues of classical polynomial inequalities, as well as direct and converse approximation theorems were proved. We gave error estimates for the weighted approximation of functions with Freud-type weights, by entire functions interpolating at finitely or infinitely many points on the real line. The error estimates involve weighted moduli of continuity corresponding to general Freud-type weights for which the density of polynomials is not always guaranteed

Schur type inequalities for multivariate polynomials on convex bodies

In this note we give sharp Schur type inequalities for multivariate polynomials with generalized Jacobi weights on arbitrary convex domains. In particular, these results yield estimates for norms of factors of multivariate polynomials

Density of Extremal Sets in Multivariate Chebyshev Approximation

AbstractThere are numerous results concerning the density of extremal sets (points of maximal deviation) in univariate Chebyshev approximation. In this note, we show that in multivariate setting this density is preserved in some weak sense

On Bernstein and Markov-Type Inequalities for Multivariate Polynomials on Convex Bodies

AbstractLet pn be a polynomial of m variables and total degree n such that ‖pn‖C(K)=1, where K⊂Rm is a convex body. In this paper we discuss some local and uniform estimates for the magnitude of grad pn under the above conditions

Homogeneous polynomial approximation on convex and star like domains

In the present paper we consider the following central problem on the approximation by homogeneous polynomials: For which 0-symmetric star like domains K ⊂ ℝd and which f ∈ C(∂ K) there exist homogeneous polynomials hn, hn+1 of degree n and n + 1, respectively, so that uniformly on ∂ K (formula presented) This question is the analogue of the Weierstrass approximation problem when polynomials of total degree are replaced by the homogeneous polynomials. A survey of various recent results on the above question is given with some relevant open problems being included, as well. © 2023, Padova University Press. All rights reserved

On the existence of optimal meshes in every convex domain on the plane

In this paper we study the so called optimal polynomial meshes for domains in K⊂Rd,d≥2. These meshes are discrete point sets Yn of cardinality cnd which have the property that (norm of matrix)p(norm of matrix)K≤A(norm of matrix)p(norm of matrix)Yn for every polynomial p of degree at most n with a constant A≫1 independent of n. It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d=2

Bernstein-Markov type inequalities and discretization of norms

In this expository paper we will give a survey of some recent results concerning discretization of uniform and integral norms of polynomials and exponential sums which are based on various new Bernstein-Markov type inequalities. © 2021, Padova University Press. All rights reserved