10,139 research outputs found
Extremalis problémák többváltozós és súlyozott polinomokra = Extremal problems for multivariate and weighted polynomials
JĂłl ismert hogy a többváltozĂłs polinomok sűrűek a d-dimenziĂłs kompakt halmazokon folytonos fĂĽggvĂ©nyek terĂ©ben. A többváltozĂłs polinomok egy fontos rĂ©szhalmaza a homogĂ©n polinomok osztálya. Igy termĂ©szetesen felmerĂĽl az a kĂ©rdĂ©s, hogy igaz-e a sűrĂĽsĂ©g a homogĂ©n polinomokra? Egy ismert sejtĂ©s szerint a konvex felĂĽleteken folytonos fĂĽggvĂ©nyek megközelĂthetĹ‘ek kĂ©t homogĂ©n polinom összegĂ©vel. A pályázat keretĂ©ben kĂ©t fontos Ăşj eredmĂ©ny szĂĽletett 1) igazoltuk a sejtĂ©st tetszĹ‘leges sima ( egyĂ©rtelmĂĽ támasz sikkal rendelkezĹ‘) konvex testeken egyenletes normában 2) igazoltuk a sejtĂ©st teljes általánosságban Lp normában Ezen kivĂĽl általánosĂtott Freud sĂşlyokra vonatkozĂł polinom-approximáciĂłs problĂ©mákat vizsgáltunk. Itt az általánosĂtás azt jelenti, hogy az eredeti Freud sĂşlyokat megszorozzuk olyan un. általánosĂtott polinomokkal, amelyeknek csak valĂłs gyökeik vannak. A klasszikus polinom-egyenlotlensĂ©gek analogonjait, valamint direkt Ă©s fordĂtott approximáciĂłs tĂ©teleket bizonyĂtottunk. HibabecslĂ©seket adtunk fĂĽggvĂ©nyek sĂşlyozott approximáciĂłjára Freud sĂşlyok esetĂ©n, olyan egĂ©sz fĂĽggvĂ©nyekkel törtĂ©no approximáciĂł esetĂ©n, amelyek vĂ©ges, ill. vĂ©gtelen sok pontban interpolálják a fĂĽggvĂ©nyt. Ezek a hibabecslĂ©sek olyan sĂşlyozott folytonossági modulusokat tartalmaznak, amelyeknĂ©l a polinom-surusĂ©g 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
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