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Geometric Analysis and Partial Differential Equations Seminar
November 20, 2014 @ 10:30 am - 11:30 pm
Prof Rodney Nillsen (University of Wollongong)
Generalised differences and multiplier operators in classical Fourier analysis
Any zeros in the multiplier of an operator impose a condition on a function for it to be in the range of the operator. But if each function in a certain family of functions satisfies such a condition, when is this family the same as the range? We consider a case where the family of functions is formed by taking finite sums of “generalised differences”, whose precise form derives from the operator. Although more general results are possible, the main case considered here is when the operator acts on a second order Sobolev space of the circle group $[0,2pi)$ and takes the form D^2+a^2I, where a is an integer. The techniques involve estimating integrals in higher dimensions over products of sets in a partition associated with the zeros of the multiplier and, perhaps, illustrates in a different context the “curse of dimension” referred to by Professor Sloan in his Colloquium last week.