The maximum distance problem and band sequences

We solve the following problem. For 1 ⩽ j, k ⩽ n and |j − k| ⩽ m, let a_jk be a given complex number with a_kj = ā_jk. We wish to find linearly independent vectors x₁,…,x_n such that 〈x_k, x_j〉 = a_jk for |j − k| ⩽ m and such that the distance from x_k to the linear span of x₁,…,x_k−1 is maximal for 2 ⩽ k ⩽ n. We construct and characterize all such sequences of vectors. Our solution leads naturally to the class of m-band sequences of vectors in an inner product space. We study these sequences and characterize their equivalence classes under unitary transformations.