By Claude Brezinski

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The Fractional Laplacian by C. Pozrikidis PDF

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Additional resources for Biorthogonality and its Applications to Numerical Analysis

Example text

For example integrability conditions on k yield to classes of nice operators. The most prominent class of operators, the Hilbert–Schmidt operators HS are deﬁned in terms of integrability conditions. Namely, an integral operator K on L2 (Rd ) is a Hilbert–Schmidt operator if k ∈ L2 (Rd × Rd ). The class of Hilbert–Schmidt operators HS has a natural inner product. Let K1 , K2 ∈ HS with kernels k1 , k2 , respectively. Then K1 , K2 HS := k1 , k2 L2 (Rd ×Rd ) (28) deﬁnes an inner product on HS. The associated Hilbert–Schmidt norm 1/2 K HS := K1 , K2 HS gives HS the structure of a Hilbert space [37].

Namely, an integral operator K on L2 (Rd ) is a Hilbert–Schmidt operator if k ∈ L2 (Rd × Rd ). The class of Hilbert–Schmidt operators HS has a natural inner product. Let K1 , K2 ∈ HS with kernels k1 , k2 , respectively. Then K1 , K2 HS := k1 , k2 L2 (Rd ×Rd ) (28) deﬁnes an inner product on HS. The associated Hilbert–Schmidt norm 1/2 K HS := K1 , K2 HS gives HS the structure of a Hilbert space [37]. Furthermore we recall that every Hilbert–Schmidt operator on HS is a compact operator on L2 (Rd ).

The obvious problem is the fact that L2 (Rd ) does not contain the system of eigenvectors of the translation operator Tx . But they can be considered as linear functionals on S0 (Rd ). This as well es several similar observations suggests to study operators on a Hilbert space via a dense subspace and its associated dual space. In our example it is actually possible to start from S0 (Rd ) and construct L2 (Rd ) as completion of S0 (Rd ) with respect to norm corresponding to the usual scalar product f, g = Rd f (t)g(t)dt.