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• W2050786039 abstract "The classical zeta function of Lerch has an analytic continuation as a distribution on the circle which seems to be very different from its usual analytic continuation; for example, the Bernoulli polynomials come out upside down. In this note functions on the circle T = R/Z are identified with periodic distributions on the line, and L2(T, C) = L2(T, C)/C is the reduced Lebesgue space of nonconstant square-integrable periodic functions. This Hilbert space has the wellknown basis e(nx) = e2f1nx n E Z-{t}, x E [0,1]. The complex powers of the Laplace operator [10] define an analytic group (-/A)of operators on L2(T, C), and if a = re s is sufficiently large, the prescription (-A) -s/2e(nx) = n -se(nx) leads to the integral representation (_A)Ys/2,0(X) = J1 Ks(x, y)>O(y) dy for these complex powers by the kernel Ks(x, y) = E n-se(nx)e(-ny). f1 0 Note that Ks(x, y) = l(s, x -y) + l(s, y -x), where l(s, x) = E e(nx)n -s tI > 1 is a quite classical function. (See [14, ?9.7, Example 2]; in particular, if x 0 Z, then it has an analytic continuation, andsince for x 0 0,(d/dx)(s, x) = 2vil(s 1, x)-it is smooth away from the origin.) Now by Schwartz's kernel theorem [9] there is an analytic function of s, taking values in the distribution on T x T which are smooth away from the diagonal, representing (_A)s/2 for all s in C. In particular, we can think of Ks as taking values in the distributions on the line smooth away from the origin. The result in this paper is a formula for this kernel modulo functions smooth at the origin. Received by the editors March 19, 1984 and, in revised form, June 11, 1984. 1980 Mathemlatics Subject Classificcation. Primary 1OH1O, 58G15, 81A19. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page" @default.
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• W2050786039 date "1985-02-01" @default.
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• W2050786039 title "Complex powers of the Laplace operator on the circle" @default.
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• W2050786039 doi "https://doi.org/10.1090/s0002-9939-1985-0784165-x" @default.
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