Titchmarsh convolution theorem

The Titchmarsh convolution theorem is named after Edward Charles Titchmarsh, a British mathematician. The theorem describes the properties of the support of the convolution of two functions.

E.C. Titchmarsh proved the following theorem in 1926:

This result, known as the Titchmarsh convolution theorem, can be restated in the following form:

This theorem essentially states that the well-known inclusion

is sharp at the boundary.

The higher-dimensional generalization in terms of the convex hull of the supports was proved by J.-L. Lions in 1951:

Above, ${\displaystyle \operatorname {c.h.} }$ denotes the convex hull of the set ${\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}$ denotes the space of distributions with compact support.

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