Zeitschrift f r Analysis und ihre Anwendungen Journal for Analysis and its Applications Volume 15 (1996), No. 1, 45-55 On some Subclasses of Nevanlinna Functions S. Hassi and H. S. V. de Snoo Abstract. For any function Q = Q(t) belonging to the class N of Nevanlinna functions, the function Qr = Q,( ) defined by Qr(t) = 1(2 belongs to N for all values of r E R U {}. The class N possesses subclasses No C N 1 , each defined by some additional asymptotic conditions. If a function Q belongs to such a subclass, then for all but one value of r E R {oo} the function Qr belongs to the same subclass and the corresponding exceptional function can be characterized (cf. [4]). In this note we introduce two subclasses N_ 2 C N 1 of No which can be described in terms of the moments of the spectral measures in the associated integral representations. We characterize the corresponding exceptional function in a purely function-theoretic way by suitably estimating certain quadratic terms. The behaviour of the exceptional function connects the subclasses N... 2 and N 1 to the classes No and N1, respectively, as studied in [4]. In operator-theoretic terms these notions have a translation in terms of Q-functions of selfadjoint extensions of a symmetric operator with defect numbers (1, 1). In this sense the exceptional function has an interpretation in terms of a generalized Friedrichs extension of the symmetric operator. Keywords: Symmetric operators, selfadjoint extensions,. Friedrichs extension, Q-functions, Nevanlinna functions AMS subject classification: Primary 47 B 25, 47 B 15, secondary 47 A 57, 47 A 55, 30 E 0 0. Introduction A scalar function Q = Q(t) is said to be a Nevanlinna function if it is holomorphic on C \ R and satisfies Q() = Q(t) and> 0 for all E C \ R. The set of all Nevanlinna functions is denoted by N. The subclass N 1 is the set of functions Q which belong to N and for which jImQ(iy) dy < oo. Similarly, the subclass No is the set of functions Q which belong to N and for which supylmQ(iy) <. 11>0 S. Hassi: Univ. Helsinki, Dep. Statistics, PL 54, 00014 Helsinki, Finland H. S. V. de Snoo: Univ. Groningen, Dep. Math., P.O. Box 800, 9700 AV Groningen, The Netherlands ISSN 0232-2064 / S 2.50 Heldermann Verlag