Vol. 29, No. 3, 1969

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Translation kernels on discrete Abelian groups

William Richard Emerson

Vol. 29 (1969), No. 3, 527–542

Let G be a compact Abelian group with discrete countable dual group Γ = Ĝ and let f L1(G) with Fourier transform F = f. If V is a finite subset of Γ we consider the operator FV on L2(V ):

(FVφ)(γ) =    F(γ − τ)φ(τ) φ ∈ L2(V ),γ ∈ V.

Then if {V n} is any suitably restricted sequence of finite subsets of Γ we show that

nli→m∞ |FVn| = nl→im∞{ max |(FVnφ,φ )|} = ∥f∥∞

where |FV | is the operator norm of FV on L2(V ) and (FV φ,φ) denotes the inner product of FV φ and φ (over V ).

This result is then translated into a statement concerning a special class of infinite matrices which generalize the classical Toeplitz matrices. We then apply these results in evaluating the norm of a special type of linear operator.

Mathematical Subject Classification
Primary: 42.51
Secondary: 47.00
Received: 14 April 1967
Revised: 22 November 1968
Published: 1 June 1969
William Richard Emerson