Projecting on Polynomial Dirac Spinors Page: 121
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Eighh Internatlonal Conference on ( Sm rg,
Geometry, ntegrablllty and Quantzation I
June 9-14, 2006, Varna, Bulgaria
Ivalio M. Miadenov and Manuel de LeOn, Editom r
SOE, Sofla 2007, pp 121-126on
PROJECTING ON POLYNOMIAL DIRAC SPINORS
Department of Mathematics, University of North Terxas, Denton, TX 76203, USA
Abstract. In this note we adapt Axler and Ramey's method of constructing
the harmonic part of a homogeneous polynomial to the Fischer decomposi-
tion associated to Dirac operators acting on polynomial spinors. The result
yields a constructive solution to a Dirichlet-like problem with polynomial
It is well-known  that any homogeneous real or complex polynomial Pk of de-
gree k = 0, , 2,.. in n 2 real variables x = (x, Xa,...,xn) admits an unique
Pk () = hk(z) + Il2Pk-2(x) (1)
where hk is a homogeneous harmonic polynomial of degree k, Pk-2 is a homoge-
neous polynomialofdegree k - 2, and, asusual, I1x = f + + + ..
In  Axler and Ramey presented an elegant, elementary way of constructing hk
from Pk, which involves only differentiation. In essence, for k > 0
clx12*pPk(D)log Ixl), if n = 2
hx() = (2)
C~ll"-2+p C)(l12-), ifn >2
(-2)k-1(k - 1)!, if n = 2
Ck l-2-n--2j), ifan> (3)
and where pkt(D) is the associated partial differential operator acting on smooth
functions defined on open subsets of Rn obtained by replacing a typical monomial
a x2 .. as , a + + ... n, = k, ofp by oxa'8z2 ..,8: "
As a by-product they obtained a speedy solution to the Dirichlet problem on the
unit bail of R" with polynomial boundary data which eliminates the use of the
impractical Poisson integral.
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Anghel, Nicolae. Projecting on Polynomial Dirac Spinors, paper, June 2006; (digital.library.unt.edu/ark:/67531/metadc161699/m1/1/: accessed November 20, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.