Suppose that m(ξ ) is a trigonometric polynomial with period 1 satisfying m(0) = 1 and |m is also an orthogonal function. A natural question, then, is whether this procedure can be extended to arbitrarily small intervals, i.e., whether for any ∈ (0, 1 2 ) there exists a finite set Z such that the orthogonality of ϕ(x) is ensured by the requirement that |m(ξ )| > 0 for |ξ | and for some ξ ∈ Z . In this paper, we show that this is true if and only if exceeds a strictly positive 0 which we derive explicitly.