By G. H. Hardy

There should be few textbooks of arithmetic as recognized as Hardy's natural arithmetic. considering its booklet in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have became first and foremost in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the elemental rules of the differential and crucial calculus, of the homes of countless sequence and of different issues regarding the idea of restrict.

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Using Eq. -. This is the system required for a Fourier series expansion. The Fourier transform requires the system {sin2n yO, cos2n Y O} which is orthogonal in the whole interval - 00 < 0 < + CI) • Note that i is an integer and thus denumerable, while y is a real number and thus nondenumerable. The system of Walsh functions orthogonal and complete in the whole interval - 00 < 0 < + CI) is denoted by {sal(,u, 0), cal(,u, O)}, where p is a real number. It will be shown later on that this system may be obtained by" stretching" sal(i, 0) and cal(i, 0) just as the system {sin2n yO, cos2nYO} can be obtained by stretching sin2n i 0 and cos2n i O.

12 and 13 for the intervals - 4 < ft < + 4 and - 3 < 0 < + 3. Black areas indicate the value + 1, white areas the value - 1 . By drawing a line parallel to the O-axis one obtains cal (ft, 0) or sal (ft, 0) as function of 0 for a certain value of ft. Vice versa, a line parallel to the ft-axis shows the values of cal(ft, 0) or sal(ft, 0) as function of ft for a certain value of O. The following additional formulas are important for computations with Walsh functions: wal(ft, 0) = wal (0, 0), 0 ~ ft < 1 cal (ft , 0) = cal(i, 0), i ~ ft < i + 1 --t ~ 0 < sal(ft, 0) = sal (i, 0), i - I < ft ~ i .

The triangular function of Fig. 32 a is expanded into a series of sine and c