Orthogonal Polynomials on the Sierpinski Gasket

Monomials

Definitions of Derivatives

The Sierpinski Gasket can be seen as the limit of a sequence of graphs Γ_m. We can define the graph Laplacian on SG as the renomalized limit of graph Laplacians on each level:

Δ u(x) = lim_{m → ∞} (3/2) 5^m Σ_{x∼y} (u(y) - u(x))

There are two derivatives on SG: the normal derivative and tangenetial derivative

∂_n u(q_i) = lim_{m → ∞} (5/3)^m (2u(q_i) -u(F^m_i q_i+1) -u(F^m_i q_i-1))

∂_T u(q_i) = lim_{m → ∞} 5^m (u(F^m₀ q_i+1) - u(F^m₀ q_i-1))

Definitions of the Monomials

The monomials are the kernels of the operator Δ^j. We let H^j be the space of functions f satistying Δ^jf =0. H^j is also called the set of polynomials of degree less than or equal to j.

For i=1,2,3 and j≥0, the monomials P_ji are defined to be the functions in H^j satisfying:

Δ^mP_ji(q₀) = δ_m,j δ_i,1
∂_nΔ^mP_ji(q₀) = δ_m,jδ_i,2
∂_TΔ^mP_ji(q₀) = δ_m,jδ_i,3

The set of monomials {P_ki | i = 1, 2, 3; and k = 0,…, j} is a basis for H^j.

Moreover, the monomials satisfy ΔP_ji = P_(j-1)i.

Graphs of Monomials

Below we plot some of the graphs of the monomials. Note that P_j1 and P_j2 are both symmetric with respect to the axis bisecting the angle at q₀. Likewise, P_j3 is antisymmetric with respect to the same axis.