On the Walsh Functions(*)
ON THE WALSH FUNCTIONS(*) BY N. J. FINE Table of Contents 1. Introduction. 372 2. The dyadic group. 376 3. Expansions of certain functions. 380 4. Fourier coefficients. 382 5. Lebesgue constants. 385 6. Convergence. 391 7. Summability. 395 8. On certain sums. 399 9. Uniqueness and localization. 402 1. Introduction. The system of orthogonal functions introduced by Rade- macher [6](2) has been the subject of a great deal of study. This system is not a complete one. Its completion was effected by Walsh [7], who studied some of its Fourier properties, such as convergence, summability, and so on. Others, notably Kaczmarz [3], Steinhaus [4], Paley [S], have studied various aspects of the Walsh system. Walsh has pointed out the great similarity be- tween this system and the trigonometric system. It is convenient, in defining the functions of the Walsh system, to follow Paley's modification. The Rademacher functions are defined by *»(*) = 1 (0 g x < 1/2), *»{«) = - 1 (1/2 á * < 1), <¡>o(%+ 1) = *o(«), *»(*) = 0o(2"x) (w = 1, 2, • • ■ ). The Walsh functions are then given by (1-2) to(x) = 1, Tpn(x) = <t>ni(x)<t>ni(x) ■ ■ ■ 4>nr(x) for » = 2"1 + 2n2+ • • • +2"r, where the integers »¿ are uniquely determined by «i+i<«¿. Walsh proves that {^„(x)} form a complete orthonormal set. Every periodic function f(x) which is integrable in the sense of Lebesgue on (0, 1) will have associated with it a (Walsh-) Fourier series (1.3) /(*) ~ Co+ ciipi(x) + crf/iix) + • • • , where the coefficients are given by Presented to the Society, August 23, 1946; received by the editors February 13, 1948.
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