Fredholm eigen value problem for general domains
Mitsuru Ozawa · Kodai Mathematical Journal · 1960
Formulation of the problem.Let D be a general planar domain and {D n } be its exhaustion in the usual sense.Let L 2 (D n ) be a class of single-valued, square integrable, analytic functions having a single-valued indefinite integral in D n .In L 2 (D n ) we shall, as usual, introduce the notion of the inner product (φ,ψ)D n by an integral φ(z)ψ(z)dτ Z9 I then L 2 (D n ) forms a complete Hubert space.In this space L 2 (D n ], there are the so-called reproducing kernel K n (z, u) and its adjoint ^-kernel l n (z, u) which satisfy the following identities: for any f(z) e L 2 (D n ),and ln(Z, U) = l n (U, Z), (l n (Z, U), l n (u, w)) Dn = K n (Z, w) -Γ n (z, ϊϋ),