## Extract

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function *f* (*x, y*) possesses a double Lebesgue integral over the square (–*π* < *π*; –*π*<*y*<*π*). Then the general term of the double Fourier series of this function is given by cos = *є* _{mn} { *a*_{mn} cos *mx* cos *ny* + *b*_{mn} sin *mx* sin *ny* + *c*_{mn} cos *mx* sin *ny* + *d*_{mn} sin *mx* cos *ny*} There *є* _{00} = ¼, *є*_{m0} = ½ (*m* > 0), *є*_{0n} = ½ (*n* > 0), *є*_{ms} = 1 (*m* > 0, *n*>0). the coefficients are given by the formulæ *a*_{mn} = 1/*π*^{2} ∫^{π} _{-π} ∫^{π} _{-π} *f* (*x, y*) cos *mx* cos *ny* *dx dy*, obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫_{ms} = Σ^{m} _{μ = 0} Σ ^{n} _{v = 0 } A_{μv} = 1/π^{2} ∫ ^{π} _{-π} ∫ ^{π} _{-π} *f* (s, t) sin(*m*+½) (*s* - *x*) sin (*n* + ½) (*t* - *y*)/2 sin ½ (*s* - *x*) 2 sin ½ (*t* - *y*) if *f* (*s*, *t*) is defined outside the original square by double periodicity, we have sub S_{ms} = 1/π^{2} ∫ ^{π} _{0} ∫ ^{π} _{0} *f* (*x* + *s*, *y* + *t*) + *f* (*x* + *s*, *y* - *t*) + *f* (*x* - *s*, *y* + *t*) + *f* (*x* - *s*, *y* - *t*) sin (*m* + ½)*s* / 2 sin ½* ^{s}* sin (

*n*+ ½)

*t*/ 2 sin ½

*t*

*ds dt*.

## Footnotes

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- Received May 31, 1924.

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