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.EQ
delim $$
.EN
.EQ
define At	% "A(" t ")" %
define Atau	% "A(" tau ")" %
define Attau	% "A(" t + tau ")" %
define CA	% C sub A (t) %
.EN
.ce
\fBCORRELATION FUNCTIONS,\fP
.ce
\fBVECTORS AND HIGH-DIMENSIONAL VECTOR SPACES,\fP
.ce
\fBORTHONORMAL BASIS VECTORS AND FOURIER SERIES\fP
.SP 2
.P
The fluctuation with time of a variable $A$ about its mean value
is described by the function $At$.
The time-correlation function for the variable 
is the classical ensemble average of the product of values of $At$ 
at times that differ by the interval $t$
.SP
.in +3
.DS
.EQ (1)
CA mark ~=~ < Attau ~ Atau >
.EN
.SP
.EQ
lineup ~=~ int ... int dp dq rho (p,q)  A(p,q;t+ tau ) A(p,q: tau )
.EN
.DE
.in
.SP
From a molecular dynamics simulation we obtain a time series of values 
of a variable,
such as a main-chain dihedral angle.
The averaging of the product $Attau Atau$ to obtain $CA$ can be over the time series.
We can view the time series as a vector.
The vector is defined by the sequence of values of the variable $A$
at the sample points of the simulation.
The dimension of the vector 
is equal to the number of sample points.
With this picture, the correlation function $CA$
is the appropriately normalized inner product (or dot product) of the two 
vectors, $Attau$ and $Atau$,
.SP
.in +3
.DS
.EQ (2)
CA ~=~ 1 over C sub norm ( Attau , ~ Atau )
.EN
.DE
.in
.SP
The vectors $Attau$ and $Atau$ are equal-size samples of the signal $At$,
with the start of the sequence defining the vector $Attau$ 
displaced by $t$ from the start of $Atau$.
.P
The inner product $( Attau , ~ Atau )$ is proportional to the projection
of the vector $Attau$ on $Atau$.
For small $t$ the vectors are near
parallel and the projection is large, approximately equal to the 
squared magnitude of $Atau$.
For large $t$, if the fluctuation in signal is high-frequency 
random noise, the vectors are orthogonal and the projection and $CA$
will be zero.
If the signal contains periodic components in addition to random noise, 
the vectors will oscillate about the perpendicular and 
the projection and $CA$ will be periodic.
.SP 2
.P
The following develops the above geometrical picture of the inner product on a
vector space by examining the two-dimensional case and 
extending it to higher dimensions and to vector spaces of functions.
The notion of orthonormal basis vectors is introduced to flesh out
the geometrical picture and to show, 
within the geometrical picture,
the relationship between the inner product, 
the Fourier series expansion of a function,
and the Fourier transform of a function.
.SP 2
I. $ bold "R" sup 2 $: a two-dimensional vector space.
.P
A vector in two-dimensions is an ordered sequence of two numbers,
.SP
.in +3
.DS
.EQ (3)
bold r ~=~ [ x ~ y ]
.EN
.DE
.in
.SP
The inner product of two vectors is the number (scalar) obtained
by componentwise multiplication of the two sequences,
.SP
.in +3
.DS
.EQ (4)
( bold r sub 1 ,~ bold r sub 2) ~=~ x sub 1 x sub 2 ~+~ y sub 1 y sub 2
.EN
.DE
.in
.SP
or by projection of one vector upon the other,
.SP
.in +3
.DS
.EQ (4a)
( bold r sub 1 ,~ bold r sub 2 ) ~=~ | bold r sub 1 | | bold r sub 2 | cos ( theta )
.EN
.DE
.in
.SP
where $theta$ is the angle between the vectors.
Equations (4) can be shown equivalent 
by expressing $cos ( theta ) $ in
terms of the magnitudes of the three sides of the triangle defined by the
two vectors.
The inner product or dot product is also written as 
$bold r sub 1 ~cdot~ bold r sub 2 $.
.P
We introduce mutually orthogonal unit vectors (orthonormal basis vectors), 
$bold i$ and $bold j$, 
.SP
.in +3
.DS
.EQ (5)
bold i ~=~ [1 ~ 0]
.EN
.br
.EQ
bold j ~=~ [0 ~ 1]
.EN
.DE
.in
.SP
We can write the vector $bold r ~=~ [ x ~ y ]$ 
as a sum of vectors,
.SP
.in +3
.DS
.EQ (6)
bold r ~=~ x bold i + y bold j
.EN
.DE
.in
.SP
It is seen that
$x = ( bold r ,~ bold i )$ and $y = ( bold r ,~ bold j)$, i.e., 
the components of $bold r$ are the projections of 
$bold r$ upon the unit basis vectors.
.SP 2
II. $ bold "R" sup n $: an $n$-dimensional vector space.
.P
The extension from two to three dimensions is obvious and familiar.
The extension to more than three dimensions, where the vector space is a
hyperspace, may be less familiar but also should be obvious.
We change notation from that for two or three dimensions
as used above
to a notation suitable to vectors with a larger number of components.
In $ bold "R" sup n $ as in $ bold "R" sup 2 $ a vector is a sequence of $n$ numbers,
.SP
.in +3
.DS
.EQ (7)
bold x ~=~ [ x sub 1 ~ x sub 2 ~...~ x sub n ]
.EN
.DE
.in
.SP
The inner product of two vectors is defined as for two dimensions, as the
componentwise product of the two sequences,
.SP
.in +3
.DS
.EQ (8)
( bold x ,~ bold y ) ~=~ sum from i=1 to n x sub i y sub i
.EN
.DE
.in
.SP
A sequence of orthonormal basis vectors for $ bold "R" sup n $ is
.SP
.in +3
.DS
.EQ (9)
bold e sub 1 ~=~ [1 ~ 0 ~ ... ~ 0]
.EN
.br
.EQ
bold e sub 2 ~=~ [0 ~ 1 ~ ... ~ 0]
.EN
.br
.EQ
~~~~ pile {. above . above .}
.EN
.br
.EQ
bold e sub n ~=~ [0 ~ 0 ~ ... ~ 1]
.EN
.DE
.in
.SP
As for two dimensions, a vector
$bold x ~=~ [ x sub 1 ~ x sub 2 ~...~ x sub n ]$
can be expressed as a sum of scaled basis vectors,
.SP
.in +3
.DS
.EQ (10)
bold x ~=~ sum from i=1 to n ( bold x ,~ bold e sub i ) bold e sub i
.EN
.DE
.in
.SP
where the inner product coefficients are the projections of the vector
upon the unit basis vectors.
.P
The geometrical picture, e.g., of basis vectors, of the inner product,
and of projection, is for higher dimensions the same as for two or
three dimensions.
The geometrical parallels serve as the basis for an intuitive 
understanding of the properties of a hyperspace.
.P
A signal sampled discretely, as in a molecular dynamics simulation, 
can be represented as a vector in $ bold "R" sup n $,
with the dimension $n$ being the size of the sample.
The picture given for $CA$ in the first two paragraphs immediately
follows from the development in this subsection.
If the first two paragraphs were not clear on first reading, perhaps they
should be reviewed now.
.P
For a molecular dynamics simulation of a protein with N atoms,
a snapshot of the trajectory taken at a sample point,
with values for 3N coordinates and 3N velocities, 
can be viewed as a vector in a 3N-dimensional hyperspace describing
the conformation or as a vector in the 6N-dimensional phase space
of statistical mechanics.
As the system evolves with time, the vector changes direction and magnitude.
The trajectory determined by the simulation is represented
by a sequence of such vectors.
The vectors define
a set of points in configuration or phase space.
The points will be clustered with varying density, but will be most dense
about one or several regions.
Note that we are thinking here as though in two or three dimensions.
Although a 3N or 6N-dimensional vector space cannot be visualized,
we note again, 
that geometrical concepts extended from two and three dimensions
provide an intuitive and comfortable understanding of the more complex systems.
.SP 2
III. $ bold "L" sup 2 ( a , b )$: the infinite-dimensional vector space 
of functions square integrable over the range $a$ to $b$.
.P
The geometrical picture extends to vector spaces of well-behaved
functions and infinite sequences of functions.
.P
The inner product is defined on $ bold "L" sup 2 ( a , b )$ as
.SP
.in +3
.DS
.EQ (11)
( bold f ,~ bold g ) ~=~ integral sub a sup b f ( t ) { g ( t ) } bar d t
.EN
.DE
.in
.SP
This expression is similar to the inner product in $bold R sup n$,
which is obtained by componentwise multiplication of the vectors.
To see this similarity, consider the definition of an integral, 
where the integrand is divided into small segments that are summed.
In $ bold "L" sup 2 ( a , b )$ as in $bold R sup n$, 
the inner product is related to the projection of
the function $f ( t )$ upon the function $g ( t )$.
The expression and picture developed for 
$CA$ in the first two paragraphs holds equally well for $Attau$ and $Atau$
being functions rather than  discrete sequences.
.P
In $ bold "L" sup 2$ as in $bold R sup n$,
it should be possible to find a complete orthonormal system of basis
vectors, $bold e sub n$.
Each basis vector in $ bold "L" sup 2$ will be itself a function,
which is different from but parallels the situation in $bold R sup n$, 
a vector space of ordered sequences,
where each basis vector is itself an ordered sequence
$[ 0 ~ ... ~ 1 ~ ... ~ 0]$.
As in $bold R sup n$,
a function can be expressed as a sum of terms, the projection of the
function upon each basis vector times the basis vector,
.SP
.in +3
.DS
.EQ (12)
bold f ~=~ sum from {n=- inf} to inf ( bold f ,~ bold e sub n ) bold e sub n
.EN
.DE
.in
.P
The archetypal orthogonal expansion of this kind is the 
Fourier series, 
for which the basis vectors are
.SP
.in +3
.DS
.EQ (13)
bold e sub n ( t ) ~=~ (2 pi ) sup {- 1 / 2} e sup {i n t}
~~~~~~~~~ - pi < t < pi
.EN
.DE
.in
.SP
$( bold e sub n ) sub {- inf} sup inf$
is a complete orthonormal sequence in $ bold "L" sup 2 ( - pi , pi )$.
Thus we can write for a real 2$pi$-periodic function
.SP
.in +3
.DS
.EQ (14a)
bold f ( t ) ~=~ (2 pi ) sup {- 1 / 2} sum from {n=- inf} 
to inf c sub n e sup {i n t}
~~~~~~~~~ n \(mo bold Z
.EN
.SP
.EQ (14b)
~~~~~~~~~~~~ c sub n mark ~=~ ( bold f ,~ bold e sub n )
.EN
.SP
.EQ
lineup ~=~ (2 pi ) sup {- 1 / 2} integral sub {- pi} sup pi 
bold f ( t ) e sup {- i n t} d t
.EN
.DE
.in
.SP
Each inner product coefficient $c sub n$ is a number which gives the
size of a contribution of frequency $n$ to the function $bold f ( t )$.
That $e sup {i n t}$ describes a contribution of frequency $n$
is apparent from the Euler formula
.SP
.in +3
.DS
.EQ (15)
e sup {i n t} ~=~ cos ( n t ) ~+~ i sin ( n t )
.EN
.DE
.in
.SP
.P
The Fourier transform is similar to the Fourier series expansion just
described.
In the series expansion the indices $n$ of the system
of basis functions are integers, ranging from $- inf$ to $inf$.
Each basis function makes a contribution of integer frequency $n$.
For the Fourier transform, we allow the "index" to be continuous, 
ranging now over the real numbers from $- inf$ to $inf$,
and we make a conforming change in notation,
replacing the index $n$ by the variable $ omega $.
We also remove the restriction on the range of $bold f ( t )$.
The relevant vector space is $ bold "L" sup 2 ( - inf , inf )$.
.P
The Fourier transform of $bold f ( t )$ is
.SP
.in +3
.DS
.EQ (16)
bold f hat ( omega ) ~=~ (2 pi ) sup {- 1 / 2} int sub {- inf}
sup inf bold f ( t ) e sup {- i omega t} d t
~~~~~~~~~ omega \(mo bold R
.EN
.DE
.in
.SP
The original function $bold f ( t )$ is recovered from
$bold f hat ( omega )$ by the inverse Fourier transform
.SP
.in +3
.DS
.EQ (17)
bold f ( t ) ~=~ (2 pi ) sup {- 1 / 2} int sub {- inf}
sup inf bold f hat ( omega ) e sup {i omega t} d omega
.EN
.DE
.in
.SP
The transform $bold f hat ( omega )$ is a vector space inner product,
projecting the function $bold f ( t )$ on the function
$(2 pi ) sup {- 1 / 2} e sup {i omega t}$.
The transform is similar to the coefficients $c sub n$ of the
Fourier series expansion, except the index has become continuous and 
can take any value.
The inverse transform, returning the function, is similar
to the summation of the Fourier series.
.P
Taking the Fourier transform of a function $bold f ( t )$ is a mapping
from the $t$ to the $ omega $ domain, 
i.e., from the time to the frequency domain.
The Fourier transform changes the basis from a function of 
time to a function of frequency.
.SK
\fBSUGGESTED READING\fP
.SP
1. N. Young, \fIAn introduction to Hilbert space\fP,
Cambridge University Press, Cambridge, 1988, 
soft\ cover, ISBN\ 0-521-33717-8, QA322.4.Y68\ 1988.
.SP
A readable but mathematical treatment of vector spaces of sequences
and functions and of linear operators.
There is a list of references on page 225.
.SP 2
2. A. Szabo and N. S. Ostlund, \fIModern quantum chemistry: Introduction
to advanced electronic structure theory\fP, 1st edition, revised,
McGraw-Hill, New York, 1989, hard\ cover, ISBN\ 0-07-062739-8,
QD462.S95\ 1989
.SP
Chapter 1 has an excellent concise review of the mathematical background 
for quantum mechanics, including: vector spaces over $bold C sup n$,
linear (matrix) algebra, basis functions, eigenvalues, extensions to 
vector spaces of functions, and linear operators.
Standard quantum mechanics texts (e.g., Schiff) also give this background.
.SP 2
3. E. Kreyszig, \fIAdvanced engineering mathematics\fP, 7th edition,
John Wiley, New York, 1993, hard\ cover, ISBN\ 0-471-55380-8,
QA401.K7\ 1993.
.SP
A text that is a good general reference.  Not mathematical.  The treatment
of linear algebra is thorough.  The treatment of vector spaces is
sketchy.
.SP 2
4. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
\fINumerical recipes in C\fP, 2nd edition,
Cambridge University Press, Cambridge, 1992,
hard\ cover, ISBN\ 0-521-43108-5, QA297.N866\ 1992.
.SP
A book of algorithms for computations, with code in C that can be
obtained on diskette.
The book covers a wide range of topics, including function minimization,
eigenvalue problems and matrix manipulations, statistical treatment of
data, differential equations, Fourier transform, etc.
Useful and comprehensible background is given for each topic, with
pointers to references.
A good book to find out what numerical methods are available for work
with a problem.
Purists and perhaps some others nitpick the code.
