**Memoirs of the American Mathematical Society**

2017;
107 pp;
Softcover

Print ISBN: 978-1-4704-2324-7

Product Code: MEMO/247/1174

List Price: $75.00

Individual Member Price: $45.00

**Electronic ISBN: 978-1-4704-3709-1
Product Code: MEMO/247/1174.E**

List Price: $75.00

Individual Member Price: $45.00

# The Mathematics of Superoscillations

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*Yakir Aharonov; Fabrizio Colombo; Irene Sabadini; Daniele C. Struppa; Jeff Tollaksen*

In the past 50 years, quantum physicists have discovered, and
experimentally demonstrated, a phenomenon which they termed
superoscillations. Aharonov and his collaborators showed that
superoscillations naturally arise when dealing with weak values, a
notion that provides a fundamentally different way to regard
measurements in quantum physics. From a mathematical point of view,
superoscillating functions are a superposition of small Fourier
components with a bounded Fourier spectrum, which result, when
appropriately summed, in a shift that can be arbitrarily large, and
well outside the spectrum.

The purpose of this work is twofold: on one hand the authors
provide a self-contained survey of the existing literature, in order
to offer a systematic mathematical approach to superoscillations; on
the other hand, they obtain some new and unexpected results, by
showing that superoscillating sequences can be seen of as solutions to
a large class of convolution equations and can therefore be treated
within the theory of analytically uniform spaces. In particular, the
authors will also discuss the persistence of the superoscillatory
behavior when superoscillating sequences are taken as initial values
of the Schrödinger equation and other equations.