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Dirichlet Kernel: Fountainhead of All Subspace Vectors

Consider the space of functions which lie in the subspace

One can say that each of these functions owes its existence to the Dirichlet kernel

 (214)

First, note that is a vector in for every . Second, note that this vector generates a set of orthonormal basis vectors for . They are generated by repeated shifts of the function along the -axis. Indeed the resulting vectors are

where

is the amount by which the function has been shifted to the right. The increment between successive shifts is evidently

the separation between the successive zeroes of in the interval . This means that, to obtain the function , the maximum of has been shifted to the location of its zero. As a consequence, note that

or more briefly,

 (215)

This is called the sifting property of the function . What is the significance of this important property? To find out compare it with the fact that the functions are orthogonal relative to the given inner product; in particular

That is, except for a normalization factor, the set of elaments forms an orthonormal basis for the subspace . Note that the property

does not depend on the inner product structure of the subspace at all. Instead, recall that this property is a manifestation of a universal property which all vector spaces have, regardless of what kind of inner product each one may or may not be endowed with. This universal property is, of course, the Duality Principle: Every vector space, in our case , has a dual vector space, which is designated by and which is the space of linear functionals on . In particular, this property expresses the duality between the given basis

for and the dual basis

for , the space of linear functionals on . A typical basis functional (dual basis element'')

is the linear map (evaluation'' map) which assigns to the vector the value of (viewed as a function) at .

By applying this linear functional to each of the basis vectors in one finds that

This duality relationship between the two bases, we recall, verifies the duality between and .

The usefulness of this evaluation'' duality is that one can use it to solve the following reconstruction problem:

Given:

• a set of samples of the function

• a basis for consisting of functions with the sifting property

Find: a set of coefficients such that

whenever .

This problem has an easy solution. Letting , and using the duality relation, one finds

Consequently,

 (216)

The amazing thing about this equation is that it not only holds for the sampled values but also for any in the interval .

How is it, that one is able to reconstruct with 100% precision on the whole interval by only knowing at the points ?

Answer: we are given the fact that the function is a vector in . We also are given that the functions , form a basis for , and that these functions have the same domain as . Equation (2.16) is a vector equation. Consequently, its reinterpretation as a function equation is also 100% accurate.

Exercise 21.2 (DIRICHELET BASIS)
Consider the ( )-dimensional space which is spanned by the O.N. basis :

Next consider the set of shifted Dirichelet kernel functions,

Show that

is a basis (Dirichelet'' basis) for . This, we recall, means that one must show that
(a)
the set is one which is linearly independent, and
(b)
the set spans , i.e. if is an element of , then one must exhibit constants such that

Next: Whittaker-Shannon Sampling Theorem: The Up: Three Applications Previous: Solution to Wave Equation   Contents   Index
Ulrich Gerlach 2010-12-09