By Edward B. Saff, Vilmos Totik (auth.)
In contemporary years approximation thought and the speculation of orthogonal polynomials have witnessed a dramatic bring up within the variety of suggestions of inauspicious and formerly untouchable difficulties. this can be as a result interplay of approximation theoretical thoughts with classical strength idea (more accurately, the idea of logarithmic potentials, that is at once concerning polynomials and to difficulties within the aircraft or at the actual line). lots of the functions are in response to an exten sion of classical logarithmic capability thought to the case while there's a weight (external box) current. The record of contemporary advancements is kind of awesome and contains: construction of the speculation of non-classical orthogonal polynomials with re spect to exponential weights; the speculation of orthogonal polynomials with admire to basic measures with compact aid; the speculation of incomplete polynomials and their common generalizations, and the speculation of multipoint Pade approximation. the recent strategy has produced lengthy sought strategies for plenty of difficulties; such a lot particularly, the Freud difficulties at the asymptotics of orthogonal polynomials with a appreciate to weights of the shape exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the matter of the precise asymptotic consistent within the rational approximation of Ixl. One objective of the current e-book is to supply a self-contained advent to the aforementioned "weighted" capability concept in addition to to its various purposes. As a side-product we will additionally absolutely improve the classical thought of logarithmic potentials.
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Extra resources for Logarithmic Potentials with External Fields
Example text
L1. L1(c) =j:. Ll(z) It - zl / exists for almost all t. Ll(zz) is finite. 29). 30) and this proves the nonnegativity of the energy integral. 31) vanishes outside a compact set. Since for t = Re icp we have for large R 1 - - = R- I Iz - tl -I/Z ( (1 - ~e-iCP) 1R - ~eicp )-I/Z R it follows that for large Rand k = 0, 1, ... o 11< e-,kcp . / -1< = -1 2rr = (_I)k 1 . L(z). 2 Minimum Principle, Dirichlet Problem 35 Since this power series vanishes for all large R, we must have f zmzm+kdfL(z) = 0 for all m, k :::: O.
O The next result, which will also be frequently used, gives a lower estimation for weighted polynomials. 6. Let w be an admissible weight and Pn(z) = zn polynomial of degree n. Then + ... be a monic "sup" [w(z)]nIPn(z)1 ::: exp(-nFw). ZESw Proof. Let w := wI sw' i;:= Sw. Then ILw = ILw is obvious from the definitions. Furthermore, let an be the discrete measure that has mass 1/ n at every zero of Pn (counting multiplicity) so that 1 n 1 IPn(z)1 UUn(z) = - l o g - - . e. 4) holds for any a with compact support.
E. on a domain of the Riemann sphere C := C U {oo}. Superharmonic functions are closely related to logarithmic potentials of positive measures. Let /L be a finite positive Borel measure of compact support. Its logarithmic potential is defined by U J1 (z) := flOg _1_ d /L(t). Iz - tl This integral is well-defined (and may equal +00), and we now show that it is superharmonic. 6. The potential UJ1 is superharmonic in C and harmonic at each point z not in the support of /L. Proof. To verify the superharmonicity of UJ1 we have to prove a) - c) with f = UJ1.
