Overview

A remarkable observation made independently by several individuals [91, 40, 64, 74, 4, 24] allows us to construct smooth orthogonal bases subordinate to arbitrary partitions of the line. The bases consist of sines or cosines multiplied by smooth, compactly supported cutoff functions, or more generally they are arbitrary periodic functions smoothly restricted to adjacent overlapping intervals. These localized trigonometric functions remain orthogonal despite the overlap, and the decomposition maps smooth functions to smooth compactly supported functions. We will describe both the bases and transformations into those bases in this chapter.

A generalization introduced in [117] of the local trigonometric transform provides an orthogonal projection onto periodic functions which also preserves smoothness. This smooth local periodization permits us to study smooth functions restricted to intervals with arbitrary smooth periodic bases, without creating any discontinuities at the endpoints or introducing any redundancy. The inverse of periodization takes a smooth periodic function and produces a smooth, compactly supported function on the line. The injection is also orthogonal, so we can generate smooth, compactly supported orthonormal bases on the line from arbitrary smooth periodic bases. In particular, we get localized exponential functions by applying the injection to complex exponentials.

The basis functions produced in this way are "windowed" in the sense that they are just the original functions multiplied by a compactly supported bump function. Hence we can control their analytic properties by choosing the bump, which we can do in an almost arbitrary manner. In particular, we can arrange...