Professor Walter Bloom

Room: S&C/3.010

Research

Harmonic analysis, Fourier analysis, universal geometry.

Projects

Convolution of measures

Bounded measures on the real line can be "multiplied" via the convolution formula

and this, together with addition, makes the space of bounded measures into a commutative Banach algebra. There are similar convolutions for measures on the half-line, many of which arise from practical problems involving classical differential equations. The Fourier analysis of these structures leads to a variety of applications in probability theory and other areas.

Trigonometric sums and Fourier series

Sinusoidal sums are the basis of Fourier series and are widely used in Engineering, in particular Power Engineering. There are many interesting properties of these, some arising just for the sake of working with nice patterns in trigonometric series, and others with very specific applications to Engineering.

Universal geometry

A recent approach to classical Euclidean geometry has been to put this on a firmer logical footing by taking a very algebraic approach. This fits in well with other geometries, including the non-Euclidean geometries of Lobachevsky, Bolyai and Gauss, and also geometry on general fields. Constructions in this area are suggested by working with mathematics software packages such as the Geometer’s Sketchpad and MAPLE.

I have several interesting projects in each of the above three areas.

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