The present book is a collection of variations on a theme which can be
summed up as follows: It is impossible for a non-zero function and its
Fourier transform to be simultaneously very small. In other words, the
approximate equalities x:::::: y and x:::::: fj cannot hold, at the same
time and with a high degree of accuracy, unless the functions x and yare
identical. Any information gained about x (in the form of a good
approximation y) has to be paid for by a corresponding loss of control
on x, and vice versa. Such is, roughly speaking, the import of the
Uncertainty Principle (or UP for short) referred to in the title ofthis
book. That principle has an unmistakable kinship with its namesake in
physics - Heisenberg's famous Uncertainty Principle - and may indeed be
regarded as providing one of mathematical interpretations for the
latter. But we mention these links with Quantum Mechanics and other
connections with physics and engineering only for their inspirational
value, and hasten to reassure the reader that at no point in this book
will he be led beyond the world of purely mathematical facts. Actually,
the portion of this world charted in our book is sufficiently vast, even
though we confine ourselves to trigonometric Fourier series and
integrals (so that "The U. P. in Fourier Analysis" might be a slightly
more appropriate title than the one we chose).