For an arbitrary Banach space formula_33 let formula_34 be a strongly continuous, uniformly bounded faithful action of formula_35 on "X". For every formula_36, consider its eigenspace formula_37. Then "X" is the closed linear span of the eigenspaces: formula_38. Assume that every eigenspace is one-dimensional and pick an element formula_39 such that formula_40. Then the system formula_41, or the same system in the Walsh-Paley numeration of the characters formula_42 is called generalized Walsh system associated with action formula_43. Classical Walsh system becomes a special case, namely, for