Some Extremal Problems for Algebraic Polynomials in Loaded Spaces

In a loaded Jacobi space with the inner product < f, g > = Gamma(alpha + beta + 2)/2(alpha+beta+1)Gamma(alpha+1)Gamma(beta+1) integral(1)(-1) fg(1-x)(alpha)(1+ x)(beta)dx+Lf(1)g(1)+Mf(-1)g(-1) (L, M >= 0) we consider the lth derivative of the algebraic polynomial Pi((r))(N)(x) = Sigma(N)(k=N-r+1) a(k)((0)) x(k) + Sigma(N-r)(j=0) a(j)x(j) (a(N)((0)) > 0) with fixed coefficients a(k)((0)). We solve the following extremal problems: Find inf < D-l[Pi((r))(N) (x)], D-l[Pi((r))(N) (x)]> (D = d/dx, 0 <= l <= N - r) and calculate extremal polynomials.