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By Singer I.

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10) (for any fixed A > for any fixed point x of the compact set K) viewed as an equality of the L2(G) elements. ) = v(lx ).. 3), and O(x, y, A) represents the sum for the series ~ B(x, y, A) = 1 -2" I: un(x)un(y) + AN/ 4I: A~2-N)/4 In(R)un(x)un(y) 00).. n=).. convergent in L2(G) with respect to the coordinates of the point y (for fixed A > 0 and x E K). 13) in which the L 2 -norm is specified in the coordinates of the point y. n=).. n=).. 15) hold, uniform with respect to x on the fixed compact set K.

6) of the following well-known estimates for the Bessel functions 13 ): 13) See, for example, Bateman and Erdelyi [9, pp. 12 and 98]. 43), 51 L = v~::; C3 C4 Ji N - 1 RN+1. 35). 46) which provides the desired estimate for 51. 45). 42), we shall need the following auxiliary assertion. 1. 48) in which the constant that sets a limit to the growth of the O-terms is independent ofR. 1. ~;;-3/81l-1/4) is valid. 6) for the Fourier coefficient, leads us to the relation Vn = Un(X) ( k ) (N-1)/2 If:(R) = Un(X) a (Il N / 2 - 3 / 4 A;;-(N/4+ 1/ 8 )) .

If we take an arbitrary sequence of positive numbers {A~} and consider, for each n = 1,2, ... 67), we can construct a FSF of the Laplace operator in a circle K; for this FSF, the preassigned sequence {A~} will be a subsequence of the fundamental values. REMARK 4. In a similar manner, based on an arbitrary preassigned sequence of nonnegative numbers {A~}, we can construct a FSF of the Laplace operator in an N-dimensional cylinder defined as the product of an interval [0 S Xl S 11"] and an (N - I)-dimensional bounded domain 8; for this FSF, the sequence {A~} is a subsequence of the fundamental values.

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