Source
Walsh (1923), “A closed set of normal orthogonal functions”, American Journal of Mathematics 45(1):5–24, LIT-300, read in full in NOTE-281.
What was actually shown
The system (§2, formulas 2–4). φ₀ ≡ 1, and each φ_{n+1} is built from φ_n on the two halves of (0, 1) by rescaling and a sign. In binary x = Σ a_i/2^i each function is (−1) raised to a sum of digits, for example φ₂^(1) = (−1)^{a₁+a₂}. The nth has n sign changes, and each takes only ±1 except at finitely many points, where it takes the average of the one-sided limits.
Closure. Each Haar function is a finite combination of Walsh functions in the same dyadic level, and conversely. Haar’s set is closed, so Walsh’s is (§2): no non-null integrable function is orthogonal to all of them.
Convergence (Thms II–VIII).
- For continuous F, the series converges uniformly when grouped in dyadic blocks of 2^{n−1} terms (Thm II), because the block sums equal Haar’s.
- For F of bounded variation, the ungrouped series converges at every point of continuity and every dyadic rational (Thm IV).
- Convergence at a point depends only on F near it (Thm V).
- For continuous F on the closed interval, the series is uniformly Cesàro summable (Thm VII); the dyadic-rational case of the pointwise statement is omitted.
- For any point there is a continuous function whose expansion diverges there (Thm VIII), by Haar’s criterion on unbounded Lebesgue constants.
- Any uniformly bounded orthonormal set obeys Riemann–Lebesgue, and Haar’s set, being unbounded, fails it (Thm VI).
Uniqueness (Thms IX–X). A series converging to 0 uniformly off finitely many points has all coefficients 0.
What could have come out otherwise. A ±1 system could have shared Haar’s pathologies, such as the failure of Riemann–Lebesgue. Being uniformly bounded and global, it shares instead the behaviour of the trigonometric system, including its divergence phenomena.
What this does not say
- That the Walsh functions are group characters. The paper never mentions groups, characters or (ℤ/2)ⁿ, never states that a product of two Walsh functions is a Walsh function, and does not mention Rademacher (NOTE-281, corrections). That they are the characters of the compact group (ℤ/2)^ℕ is a later identification, an instance of THEORY-645. For the character view of the Boolean cube, cite LIT-346.
- The Paley or Hadamard ordering. The order is by number of sign changes.
- A discrete or multidimensional version. The paper works on the interval only.
- Anything about predicates or Boolean functions. The owner’s map row 7, “predicate harmonics”, needs the character theory, which this paper does not supply.
Connections
- THEORY-019 uses Walsh characters as an example of pieces inside a multiplet. THEORY-645 is the duality theorem of which their completeness is a case.