The smoothest average: Dirichlet, Fejér and Chebyshev

Abstract

We are interested in the ‘smoothest’ averaging that can be achieved by convolving functions f ∈ ℓ 2 ( Z ) with an averaging function u. More precisely, suppose u : { − n , … , n } → R is a symmetric function normalized to ∑ k = − n n u ( k ) = 1 . We show that every convolution operator is not-too-smooth, in the sense that sup f ∈ ℓ 2 ( Z ) ∥ ∇ ( f ∗ u ) ∥ ℓ 2 ( Z ) ∥ f ∥ ℓ 2 ⩾ 2 2 n + 1 , and we show that equality holds if and only if u is constant on the interval { − n , … , n } . In the setting where smoothness is measured by the ℓ 2 -norm of the discrete second derivative and we further restrict our attention to functions u with nonnegative Fourier transform, we establish the inequality sup f ∈ ℓ 2 ( Z ) ∥ Δ ( f ∗ u ) ∥ ℓ 2 ( Z ) ∥ f ∥ ℓ 2 ( Z ) ⩾ 4 ( n + 1 ) 2 , with equality if and only if u is the triangle function u ( k ) = ( n + 1 − | k | ) / ( n + 1 ) 2 . We also discuss a continuous analogue and several open problems.