Acta Mathematica Academiae Scientiarum Hungaricae 64. (1994)
1994 / 1. szám - Horváth M.: Local uniform convergence of the eigenfunction expansion associated with the Laplace operator. I
2 M. HORVÁTH with some h € Lp(Tl.N) and we define the norm This extends the notion of Soboleff spaces W" for nonintegral a. The main results of this paper are the following statements published without proof in [12]. Theorem 1. Suppose supp / C П is compact and of the expansion of f tend to f locally uniformly in П. THEOREM 2 (localization principle). Let supp / C П be compact and suppose that for some other domain Oo C П we have Then the expansion of f tends to zero locally uniformly in По-Theorem 3 (absolute convergence). Suppose again that supp / С П and /eXp(HN), a > y, ap > N, p^l. Then the expansion of f converges absolutely and locally uniformly in П. Remarks 1. The above theorems extend some results of E. C. Titchmarsh [1] and V. A. Il’in [5], [6]. They investigated the same problems for A, ^ 0. For the case of arbitrary complex eigenvalues these theorems were obtained for N = 3 by I. Joó [8], [9], [10]. Earlier the case N — 1 with complex eigenvalues and for the Schrödinger operator was obtained by I.Joó and V. Komornik in [7]. 2. Using the ideas of A. Bogmér [11] we can generalize the above theorems for the case of higher order eigenfunctions. 3. We can give examples of Riesz bases (Ui) with complex eigenvalues A; (even with sup \u[\ = oo) if П = (0,27r)^ by the “direct product” of onedimensional exponential bases, see e.g. [14]. f e L“(Rn), asti, Then the partial sums (3) Sß(f,x):= ftui(x), f, pí<u := / fvi, J fi p > 0 ap > N, p ^ 1. f e L%{Rn), f\Qo = 0. Acta Mathemaiica Hungarica 64, 1994