Harmonic Analysis and Function Spaces
Overview of Research
I use a variety of methods to study connections between properties of functions and their Fourier transforms. For instance, in my 1997 paper in the Tohoku Mathematics Journal, I consider integrable functions on the real line with nonnegative transforms, and show that such functions are square-integrable in some neighbourhood of the origin if and only if their transforms are locally integrable and globally square-integrable. That leads to new proofs that such functions need not be globally square-integrable, and these methods extend to some situations where the conclusions are new.
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 two proofs of particular fact, one by a method in the folklore 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 new ways, because those methods led to new results of great interest to me.
In those cases and some others, my work was motivated by specific unsolved problems. In other cases, I initially sought a better understanding of a known proof, and that led to new proofs and new conclusions.
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