The $L^p$ convergence of Fourier series on triangular domains

Abstract

We prove $L^p$ norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lamé's Theorem.

Figures (5)

• Figure 1: Reflecting $T$ into the square $[0,1]^2$ along the diagonal $y=1-x$.
• Figure 2: The equilateral triangle $T$ (shaded), the right-angled triangle $T_1$ and six congruent copies $T_i$ arranged into the rectangle $R=[0,\sqrt{3}]\times[0,1]$. The "$\pm$" signs indicate a symmetric or antisymmetric reflection, respectively, in the definition of $\mathcal{P}_a$. Assuming zero boundary conditions on $T_1$, the extension by $\mathcal{P}_a$ vanishes on all lines draw in $R$ and its boundary. See Prager1998.
• Figure 3: The three regions in the lattice $[1,16]\times[1,16]$. The lines correspond to the "degenerate cases" $m=n$ and $m=3n$ in which $u_{m,n}\equiv 0$, and thus the Fourier coefficients vanish. Note that all points where $m$ and $n$ have opposite parity will also be excluded.
• Figure 4: The sign arrangements for the symmetric prolongation $\mathcal{P}_s$. See Prager1998.
• Figure 5: Limitations of the triangle-to-rectangle constructions.

Theorems & Definitions (23)

• Lemma 1
• proof
• Corollary 2
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Lemma 5
• Lemma 6