Due April 27
- converges for a.e. x.
- for a.e. .
- The operator is a Fourier multiplier operator with multiplier .
Let be open, and be the subspace of holomorphic functions on .
- is a closed subspace of .
- If is an orthonormal basis of , then for , where is the distance from z to the complement of .
- The sum converges absolutely for , and is independent of the orthonormal basis .
The function is called the Bergman kernel.
- If T is the linear transformation on defined by , , then T is the orthogonal projection of onto .
Consider the definitions of the previous problem with the unit disc.
- if and only if
- The sequence is an orthonormal basis for .
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