On the Number of Fixed Points of the Reed-Muller-Fourier Transform

On Eigenvectors of the Pascal and Reed-Muller-Fourier Transforms

“…After presenting the required definitions and tools in Section 2, we will prove in Section 3 that if h is odd and λ ∈ Z h satisfies λ 2 = 1, then the number of eigenvectors corresponding to the eigenvalue λ of an arbitrary triangular self-inverse matrix S ∈ Z N ×N h depends only on the diagonal entries of S (Theorem 3.1). This result already proves Conjecture 1.1 for odd h. Let us add that this case was also settled in [18] using a different method. The results of [18] also indicate that the space of fixed points has a basis, which is not true for arbitrary subspaces of Z N h (see Example 2.1).…”

“…The results of [18] also indicate that the space of fixed points has a basis, which is not true for arbitrary subspaces of Z N h (see Example 2.1). The proof presented here does not provide the existence of a basis, but it is simpler and more general than the proof in [18].…”

“…This certainly includes the cases λ = 1 (fixed points) and λ = −1, but in general there might be more such eigenvalues (for example, if h = 12, then λ = 5 and λ = 7 also satisfy λ 2 = 1). It was proved in [18] that if h is odd, then Z h n h has a basis consisting of eigenvectors of T ⊗n h corresponding to the eigenvalues 1 and −1. If h is a prime (i.e., if Z h is a field), then this implies that there are no other eigenvalues.…”

“…Next, we will determine the number of eigenvectors of P N corresponding to eigenvalues λ ∈ Z h with λ 2 = 1 (note that Theorem 4.1, the main result of this section, overlaps with Theorem 3.1 if h is odd). Since T h = −P h , this includes as a special case the results of [18], where one-variable eigenfunctions of the Reed-Muller-Fourier transform were considered with the eigenvalues ±1. An elimination procedure was used in [18], but its correctness was not rigorously proved (although the patterns of binomial coefficients appearing in the matrices were clear enough).…”

“…Since T h = −P h , this includes as a special case the results of [18], where one-variable eigenfunctions of the Reed-Muller-Fourier transform were considered with the eigenvalues ±1. An elimination procedure was used in [18], but its correctness was not rigorously proved (although the patterns of binomial coefficients appearing in the matrices were clear enough). Here we provide a proof, and instead of a step-by-step procedure, we do the elimination at once, by multiplying by a suitable invertible matrix.…”