On Walsh-fourier Series^)
نویسندگان
چکیده
Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine exhibited some of the basic similarities and differences between the trigonometric orthonormal system and the Walsh system. He identified the Walsh functions with the full set of characters of the dyadic group G. Contemporary with the work of Fine and somewhat more general is the work of ._ i Presented to the Society, April 26, 1952 under the title Theorems on series of Walsh functions; received by the editors September 13, 1954 and, in revised form, April 2, 1956 and May 3, 1956. C) This work, supported by the Office oi Naval Research, forms the second part of a dissertation submitted in June, 1953 in partial fulfillment of the requirements for the Ph.D. degree at the University of Chicago. The author wishes to express his gratitude to Professor Antoni Zygmund for his valuable suggestions and encouragement. The referee also has been most helpful with suggestions for shortening proofs of several theorems, particularly those of §5. (2) The numbers in brackets refer to the bibliography at the end of the paper. (3) Hereinafter "W.F.S." will denote "Walsh-Fourier series."
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تاریخ انتشار 2010