Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

When f ðtÞ and gðtÞ are continuous functions in the vector space C½a; b& with pointwise addition and scalar multiplication, the inner product ð f ; gÞ is given by

a

p

The Euclidian norm of a vector u is ðu; uÞ, and is denoted by juj. The vectors u and v are orthogonal if ðu; vÞ ¼ 0. In the space Fn, a more general norm is defined by

where 1 p < 1. p ¼ 2 gives the Euclidian norm.

Below we discuss important properties of inner-product vector spaces in general, and some inner-product vector spaces in particular.

Distance Measure (Metric)

The distance dðu; vÞ between two vectors u and v shows how similar the two vectors are. Inner-product vector spaces are also metric spaces, the metric being dðu; vÞ. The distance dðu; vÞ can be defined in many ways provided that it satisfies the following:

1.dðu; vÞ 0

2.dðu; uÞ ¼ 0

3.dðu; vÞ ¼ dðv; uÞ

4.d2ðu; vÞ d2ðu; wÞ þ d2ðw; vÞ

The last relation is usually called the Schwarz inequality or the triangular inequality. The Euclidian distance between two vectors x and y is given by

It is observed that the norm of a vector u is simply dðu; hÞ, h being the null vector. Even though the Euclidian distance shows the similarity between u and v, its magnitude depends on the norms of u and v. In order to remove this dependence, it

can be normalized by dividing it by jujjvj.

Examples of Inner-Product Vector Spaces

Two most common inner product spaces are the set R of real numbers and the set C of complex numbers, with their natural metric dðu; vÞ being the Euclidian distance between u and v.

388 APPENDIX B: LINEAR VECTOR SPACES

Cn and Rn are the n-dimensional vector spaces over C and R, respectively. The

with componentwise addition and scalar multiplication, and with inner product given by

X1

n¼1

where v ¼ ½vn&. Componentwise addition means the following: if w ¼ u þ v, then

Scalar multiplication means the following: if w ¼ lu, l a scalar, then

L2ða; bÞ denotes the vector space of continuous functions over C[a,b] (complex values in the interval from a to b) with the inner product

Angle

The angle y between two vectors x and y is defined by

y also shows the similarity between u and v. y equal to 0 shows that u and v are the same. y equal to p=2 shows that u and v are orthogonal.

Cauchy–Schwarz Inequality

The inner product of two vectors u and v satisfy

with equality iff x and y are linearly dependent (x ¼ ay, a a scalar).

Orthogonality

Let U ¼ ½u0; u1; . . . ; uN 1& be a set (sequence) of vectors in an inner-product space S. U is an orthogonal set if ðuk; u‘Þ ¼ 0 for k ¼6 ‘. U is orthonormal if

ðuk; u‘Þ ¼ dk‘.

EXAMPLE B.5 Let the inner-product space be L2ð0; 1Þ. In this space, the functions

ukðtÞ ¼ cosð2pktÞ

form an orthogonal sequence.

B.4 HILBERT SPACES

The theory of Hilbert spaces is the most useful and well-developed generalization of the theory of finite-dimensional inner-product vector spaces to include infinitedimensional inner-product vector spaces. In order to be able to discuss Hilbert spaces, the concept of completeness is needed.

Consider a metric space M with the metric d. A sequence [uk] in M is a Cauchy sequence if, for every e > 0 there exists an integer k0 such that dðuk; x‘Þ < e for k; ‘ > k0. M is a complete metric space if every Cauchy sequence in M converges to a limit in M. An example of a Cauchy sequence is the Nth partial sum in a Fourier series expansion (see Example B.5 below).

A Hilbert space is an inner-product space which is a complete metric space. For example, Cn, Rn are Hilbert spaces since they are complete.

Inner-product spaces without the requirement of completeness are sometimes called pre-Hilbert spaces.

EXAMPLE B.5 Consider a periodic signal uðtÞ with period T. The basic period can be taken to be T=2 t T=2. The Fourier series representation of uðtÞ 2 L2ð T=2; T=2Þ is given by

p Tð=2

¼2=T uðtÞ sinð2pkFstÞdt

T=2

The Hilbert space of interest in this case is periodic functions which are squareintegrable over one period. The vectors consist of all such periodic functions. The Cauchy sequences of interest are the Nth partial sums in Eq. (B.5-1). The limit of any such sequence is xðtÞ.