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LIT-300: A closed set of normal orthogonal functions

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StatusActive — read in full 2026-09-29 (NOTE-281); worth reading as the primary source of the Walsh system, and short. For Walsh functions *as characters of the Cantor group / Boolean cube* and the Walsh–Hadamard/Boolean-Fourier theory the map needs, it is superseded by later work (Paley 1932 for the Rademacher-product ordering; Fine 1949 for the dyadic-group view; O'Donnell, LIT-346, for Boolean Fourier analysis).
Version2
Tagsmathematics, information-theory
Cited as “paper” byNOTE-281 — A closed set of normal orthogonal functions
Cited as “source” byTHEORY-620 — Walsh's 1923 functions are a closed orthonormal set on (0, 1) of ±1-valued step functions, each a parity of finitely many binary digits, whose expansions behave like Fourier series: grouped uniform convergence for continuous functions, convergence under bounded variation, localisation, Cesàro summability, and divergence at any chosen point for some continuous function

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J. L. Walsh (1923), American Journal of Mathematics 45(1):5–24 — DOI-10.2307/2387224

Standing in the record

Registered on 2026-09-29 at the owner’s request, because the owner’s prior-art novelty map (a private working document on concepts as projectors, the linear representation hypothesis and their neighbours) cites it: row 7: Boolean-Fourier (characters of the Boolean cube) behind predicate harmonics. It is seeded from metadata and published descriptions and has not been read here.

Identification notes: Not the nucleation LIT-209 (a different Walsh, on Kaepernick).

It was filed Deferred, unread. NOTE-281 is the close reading of 2026-09-29, and it placed the work: Active — worth reading as the primary source of the Walsh system, and short. For Walsh functions as characters of the Cantor group / Boolean cube and the Walsh–Hadamard/Boolean-Fourier theory the map needs, it is superseded by later work (Paley 1932 for the Rademacher-product ordering; Fine 1949 for the dyadic-group view; O’Donnell, LIT-346, for Boolean Fourier analysis).

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