Research Interests
(of John J.F. Fournier)

Main Areas
Harmonic Analysis and Function Spaces

Overview of Research
I use various methods to study connections between properties of functions and their Fourier transforms. For instance, in my 1997 paper in the Tohoku Mathematics Journal, I considered integrable functions on the real line with nonnegative transforms, and showed that such functions are square-integrable in some neighbourhood of the origin if and only if suitable local norms of their transforms form a square-summable sequence. That led to new proofs that such functions need not be globally square-integrable, and these methods extended to some situations where that conclusion was new.

More recent work along those lines is posted on in a joint paper with Walter Bloom and Michael Leinert.

Sometimes my own methods do not satisfy me, and I return to a key instance where they worked, and reprove those conclusions in a different way that extends to new situations. My paper in the Pacific Journal of Mathematics in 1969 contains three proofs of a particular fact, two by known methods and one due to me; the latter allowed me to also prove some new results. In the Proceedings of the American Mathematical Society in 1974 and in Arkiv for Mathematik in 1979, I reproved the same particular fact in other new ways, because those methods led to new results of great interest to me.

More recently, I returned to these issues once again, and was able to apply the method that was new in my thesis to other questions. See my papers the missing proof, noncommutative Khintchine and discrete Fourier restriction.

Teaching | Research Interests | Publications and Preprints
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