sequencesalgebra

Sequencesalgebra is a term used to describe algebraic structures whose elements are sequences indexed by natural numbers (or integers) and that are closed under addition and a chosen multiplication. Depending on the product, a sequencesalgebra can behave differently: the most common options are coordinate-wise (Hadamard) multiplication or a convolution-type product. The choice of product, together with a compatible topology or norm, determines properties such as commutativity, associativity, and the existence of a multiplicative identity.

Common concrete instances arise in functional analysis. The space l-infinity, the set of all bounded sequences, forms a commutative Banach algebra under pointwise operations, with the sup norm. Its closed subspace c0, of sequences tending to zero, is a natural closed ideal. On the other hand, the convolution algebras l1, consisting of summable sequences with (a*b)_n = sum_k a_k b_{n-k} (defined with appropriate indexing), form unital Banach algebras with identity given by the sequence δ0. The finite-support sequences c00 form a dense subalgebra of these convolution algebras.

These algebras are closely linked to generating functions: a sequence (a_n) corresponds to a formal power series sum a_n z^n, and convolution of sequences corresponds to the product of generating functions. In harmonic analysis, Fourier transforms relate l1(Z) under convolution to multiplication of continuous functions on the unit circle.

Shifts and related operators provide a bridge to operator theory: the unilateral shift on l2 gives rise to Toeplitz-type algebras when combined with sequence spaces. Sequencesalgebras appear across functional analysis, operator theory, signal processing, and dynamical systems, where their algebraic and topological structures illuminate both abstract theory and applications.