Periodic harmonic functions on lattices and Chebyshev polynomials

Lattices and Periodic Functions

. To exclude functions, as in Example L.2.b, that are constant in some direction, it suffices to require that 0 be an isolated point of Pf . That is, to require that there be a number r > 0 such that every nonzero γ ∈ Pf obeys |γ| ≥ r. Proposition L.3 If P is an additive subgroup of IR and 0 is an isolated point of P, then there are d ≤ d and independent vectors γ1, · · · , γd′ ∈ IR d such that...

Lattices and Periodic Functions

. To exclude functions, as in Example L.2.b, that are constant in some direction, it suffices to require that 0 be an isolated point of Pf . That is, to require that there be a number r > 0 such that every nonzero γ ∈ Pf obeys |γ| ≥ r. Proposition L.3 If P is an additive subgroup of IR and 0 is an isolated point of P, then there are d ≤ d and independent vectors γ1, · · · , γd′ ∈ IR d such that...

Periodic harmonic functions on lattices and points count in positive characteristic

Generating functions of Chebyshev-like polynomials

In this short note, we give simple proofs of several results and conjectures formulated by Stolarsky and Tran concerning generating functions of some families of Chebyshev-like polynomials.

On integer Chebyshev polynomials

We are concerned with the problem of minimizing the supremum norm on [0, 1] of a nonzero polynomial of degree at most n with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and impr...