On formal integration of double trigonometric series
M. J. Kohn · Illinois Journal of Mathematics · 1978
We will be working in two dimensional Euclidean space.We denote points of E2 by x (x t, x2)= td and integral lattice points by n (n t, n2).We set [x[ (x2 + xz) /2 and n x= nixi + n2x2.By a sum ' we mean l,I o.Let (1.1) T= E cn ein'x n Z2 be a double trigonometric series which is circularly summable at Xo to finite sum s.Let T* be the series obtained by formally integrating T once with respect to x and once with respect to x2"We are interested in proving a theorem of"Riemann type" for T*.That is, we want to give conditions on the coefficients of T and on the order of summabi- lity of T which will insure that T* converges at Xo to a function F(x) which has, in some sense, at Xo a "second symmetric derivative" with value s.We define, to this end, the idea of a symmetric derivative of a function F(x) defined in a neighborhood of Xo E 2 by expanding a weighted circular mean of F(x), taken about the circle Ix Xol t, in a Taylor's series ofeven powers of t.This definition may be thought of as a two dimensional analogue of the formula (1.2) from [8, vol.2, p. 59].When the proper weighted circular mean is chosen, we are able to apply it to T* to prove a two dimensional analogue of results from [8, vol. 1, p. 320].2. We make the following definition.Let f(0) be defined for 0 e [0, 2n] such that f(0 + n) f(0).Let F(x) be defined in a neighborhood of Xo e E2 and integrable over each circle Ix-Xol t, for small.Let 2r be an even, positive integer.DEFINITION.F has, at Xo, a 2rth f-derivative with value a2r if (2.1) F(xo + te')f(O) dO a2 2 a2r =ao+2--+'''+22(r+ 1)!(r- as O.+