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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 1. Topological, metric, functional, and vector spaces
E
f
51
____________________________________________________________________________
3
for any vectors
that
, ∈ac
.
0×=aa
But, the right-hand side of (1.6.19) at
. Equation (1.6.20) immediately follows from (1.6.17) revealing
is
=ba
and the result of the multiplications can be not a zero vector. To demonstrate this, we apply
formula (1.6.18) to (1.6.21)
where
hand side of (1.6.22) does not vanish provided
The vector product obeys the following property associated with a non-degenerate matrix
D.
transformation
where
Proposition 1.6.4
Let a and b be two collinear vectors, then
××aac
()
××= θ θ
aac ac n
()
θ
ac n
sin
()
11
is a unit vector that is orthogonal to vectors a and
n
2
2
(1.6.21)
sin sin
() ( )
122
π
/2
is not collinear with a.
c
W :
⋅× ⋅ = ⋅×Wa Wb WW a b
()() () ()
T−
W is the transposed inverse matrix.
det
T−
(1.6.22)
. It is clear that the right
n
1
(1.6.23)
Proof
The proof flows out from Remark 1.6.1.B. Indeed, for collinear vectors the right-hand side of
formula (1.6.18) ensures vanishing of the vector product, since angle
vanishes.
1.6.3. Bilinear and polylinear forms, tensors,
Definitions 1.6.6 (Polylinear forms)
Let
A.
real valued function)
both arguments.
and F be two (real) finite dimensional vector spaces, the bilinear form (or bilinear
(1.6.24)
0×=ab
between these vectors
θ
and tensor spaces
is a mapping from E and F to that is linear with respect to
Chapter 1. Topological, metric, functional, and vector spaces
E
f
p
E
j
E
E
E
p
E
E
p
E
E
p
E
p
52
____________________________________________________________________________
Similarly to bilinear form a polylinear form can be defined. Let
B.
dimensional spaces, then function
linear with respect to any of its arguments:
;...; ; ; ;...;
f
aa a ba a
()
11 1
α+β =
jjjj p
−+
aaaa a aaba a
;...; ; ; ;...; ;...; ; ; ;...;
ff
α+β
()()
11 1 11 1
jj p j jj p
−+ −+
from
,1,...
kp=
k
is called polylinear, if it is
E××
to
...
1
be real finite
(1.6.25)
where
kk
kp∈=a
,1,...,
Proposition 1.6.5
If f is a polylinear form, then
Proof
The proof immediately follows from definition (1.6.25) taking α and
Remark 1.6.2
In Definition 1.6.6.A vector spaces E and F need not have equal dimensions, similarly in
Definition 1.6.6.B vector spaces
applications these spaces have usually the same dimension; or even more often the spaces
,1,...
kp=
k
can simply coincide.
Definition 1.6.7 (Tensor space)
; and α and β are real numbers.
;...; ; ; ;...; 0
f
()
111
jj p
−+
,1,...
kp=
k
can have different dimensions. However, in
(1.6.26)
=aa0a a
zeros.
β
Tensor space of tensors of
p
∏
kp
k
1
=
denoted by
Remarks 1.6.3
A. Elements of the tensor space
kk
....
EE
≡××
1
p
⊗
k
1
=
kp∈=x
, 1,...,
that is vanished by all polylinear forms f. Such a tensor space is
or
k
....
1
. Elements
-th rank is a subspace (more rigorously a factor space) in
.
E⊗⊗
p
are denoted by
⊗
k
1
=
k
....
⊗⊗xx
1
are called tensors of p-th rank.
p
⊗ x
=
k
or
k
1
....
⊗⊗xx
1
, where
Chapter 1. Topological, metric, functional, and vector spaces
j
j
E
j
p
p
E
E
E
p
p
53
____________________________________________________________________________
According to Definitions 1.6.6.B and 1.6.7 the following relation takes place
B.
... ...
⊗⊗ ⊗α +β ⊗ ⊗⊗ =
aaaba a
11 1
()
jjjj p
−+
C.
Suppose that each of the vector spaces
vector basis
-th rank. Each of the vectors
Then, tensor
Definition 1.6.8
be a finite dimensional real vector space and
Let
real valued functions defined on
=⊗⊗xa a
1
number
...
; 1,... ; 1,...,
()
j
k
...p⊗⊗aa
1
and
α⊗⊗ ⊗ ⊗ ⊗⊗ +
... ...
aaaa a
11 1
β⊗⊗ ⊗ ⊗ ⊗⊗
... ...
aaba a
11 1
−+
nk p==e
can be represented in terms of the basis vectors
a
k
n
=α
ae. (1.6.28)
kjj
()()
∑
1
j=
jj p
−+
jj p
has the same dimension n and is equipped with
k
. Let
kk
...p⊗⊗aa
1
(1.6.27)
be a corresponding tensor of
admits following representation in terms of basis vectors
p
n
...
⊗⊗ = α ⊗
aa e
1
∑∏
() ()
11
jk
==
), then the full scalar product
′′ ′
∈⊗⊗xa a
...
1
belonging to
p
jj
kk
1
k
=
′
be a space of all real linear forms (linear
(1.6.29)
,′xx
p
∏
kE=
and
k
1
p
∏
kE=
′
respectively, is a real
k
1
of two tensors
Remark 1.6.4
Tensors of the second rank (i.e. tensors having two indices) can be associated with matrices. In
physical applications tensors of higher rank can also appear. For example, in theory of elasticity
of anisotropic bodies tensors of the fourth rank specify elastic properties of anisotropic media.
p
′′
=
,,
xx a a
∑
k =
1
. (1.6.30)
kk
Chapter 1. Topological, metric, functional, and vector spaces
54
____________________________________________________________________________
Bibliography to Chapter 1
ABRAMOVITZ M. AND STEGUN I.A.
Handbook of Mathematical Functions. 10th Edition. National Bureau of Standards, 1972.
OURBAKI N.
B
Elements of Mathematics. General Topology. Chapters 1 – 4. Springer, N.Y., 1989. ISBN-13:
978-0387193748
OURBAKI N.
B
Elements of Mathematics. General Topology. Chapters 5 – 10. Springer, N.Y., 1998. ISBN-13:
978-3540645634
OURBAKI N.
B
Elements of Mathematics. Algebra. Chapters 1 – 3. Springer, N.Y., 1998. ISBN-13: 9783540642435
OURBAKI N.
B
Elements of Mathematics. Topological Vector Spaces. Springer, N.Y., 2002. ISBN-13: 9783540423386
OURBAKI N.
B
Elements of Mathematics. Functions of a Real Variable. Springer, N.Y., 2003. ISBN-13: 9783540653400
ROWN J. AND CHURCHILL R.
B
Complex Variables and Applications. McGraw-Hill Science, N.Y., 2008. ISBN-13: 9780073051949
OPSON E.T.
C
Asymptotic Expansions. Cambridge University Press, N.Y., 2004. ISBN-13: 978-0521604826
DWARDS R.E.
E
Functional Analysis. Theory and Applications. Dover Publications, N.Y., 1995. ISBN-13: 9780486681436
RDELYI A.
E
Asymptotic Expansions. Dover Publications, N.Y., 2010. ISBN-13: 978-0486603186
ÖRMANDER L.
H
The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier
Analysis. Springer, N.Y., 2003. ISBN-13: 978-3540006626
UMPHREYS T.P.
H
A Reference Guide to Vector Algebra. Jain Pub Co. 2008. ISBN-13: 978- 0875730950
ORN G.A. AND KORN T.M.
K
Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for
Reference and Review. Dover Publications, N.Y., 2000. ISBN-13: 978-0486411477
Chapter 1. Topological, metric, functional, and vector spaces
55
____________________________________________________________________________
S
CHWARTZ L.
Mathematics for the Physical Sciences. Dover Publications, N.Y., 2008. ISBN-13: 9780486466620
HILOV G.E. AND GELFAND I.M.
S
Generalized Functions. Vol.1. Properties and Operations. Academic Press, 1964.
HILOV G.E. AND GELFAND I.M.
S
Generalized Functions. Vol.2. Spaces of Fundamental and Generalized Functions. Academic
Press, 1968.
LATER L.J.
S
Generalized Hypergeometric Functions. Cambridge University Press, 2008. ISBN-13: 9780521090612
TEIN E.M.
S
Singular Integrals and Differentiability Properties of Functions. Princeton University Press.
1971. ISBN-13: 978-0691080796
TEIN E.M. AND SHAKARCHI R.
S
Fourier Analysis: An Introduction. Princeton University Press. 2003. ISBN-13: 9780691113845
ITCHMARSH E.C.
T
Theory of Functions. Oxford University Press, N.Y., 1976. ISBN-13: 978-0198533498
UNSCH D.
W
Complex Variables with Applications (3rd Edition), Addison Wesley, N.Y., 2004. ISBN-13:
978-0201756098
Chapter 2.
57
Equation Chapter 2 Section 0
Fourier
series, wavelets, and integral
transforms
In this chapter we present a brief outline of Fourier series, wavelet transforms,
Fourier, Laplace, Mellin, and some other integral transforms. The reader familiar
with these topics can easily pass to the subsequent chapters.
2.1. Equation Chapter 2 Section 1 Fourier
series
The basic properties of Fourier series are discussed in many textbooks, among which
the classical Bochner and Chandrasekharan (1949) and Titchmarsh (1962) remain
possibly the best. Some of the newer sources for the Fourier analysis are manuscripts
by Dyke (2001), Bhatia (2004), and Jackson (2004).
2.1.1. Basic definitions
Definition 2.1.1 (Periodic function)
A function
for every
t
of one (real) variable is called periodic, if there is a real positive p , such that
()f t
()()f tp f t+=
. (2.1.1)
Chapter 2. Fourier series, wavelets, and integral transforms
p
f
f
f
58
_____________________________________________________________________________
The number
minimum one, then such a period is called the fundamental (or main) period.
Remark 2.1.1
Some periodic functions can have no fundamental period, for example a constant function
is called a period of the function. If among other possible periods there is a
has no fundamental period, since any positive number is its period.
Definition 2.1.2 (Fourier series)
Fourier series for a given integrable and periodic function
with coefficients
In (2.1.3) the sign
to function
a (called also Euler, Euler-Fourier, or Fourier coefficients) defined by
k
cf ikpd
=τπττ
k
∼ reflects the fact that the formal series in the right-hand side are connected
by (2.1.4). Exponential functions
()ft
(2.1.4) are called basic or fundamental functions. Substituting expression (2.1.4) for the
coefficients into series (2.1.3), we can have another form of the Fourier series, which combines
both of the latter:
()ftc=
∞
( ) exp(2 / )
tciktp
∼
∑
k
k
=−∞
tp
+
0
1
p
()exp(2 / )
∫
t
0
(2.1.2)
π
, (2.1.3)
exp(2 / )ikt pπ
is
()ft
. (2.1.4)
appearing in (2.1.3) and
Remark 2.1.2
Recalling that
we can rewrite series (2.1.3) in terms of trigonometric functions:
with the Euler coefficients
tp
+
∞
tf d
∼ . (2.1.5)
() ( )exp
∑
k
=−∞
0
ττ
∫
pp
t
0
2( )1
ik t
⎛ πτ+⎞
⎜⎟
⎝⎠
exp(2 / ) cos(2 / ) sin(2 / )i kt p kt p i kt pπ=π+π, (2.1.6)
∞
() cos(2 / ) sin(2 / )
t a a ktp b ktp
+π+π
∼
0
()
∑
kk
1
k
=
(2.1.7)
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
59
_____________________________________________________________________________
tp
+
0
1
afd
=ττ
0
p
2
afikpd
=τπττ
k
p
2
bfkpd
=τπττ
k
p
()
∫
t
0
+
tp
0
()cos(2 / )
∫
t
0
+
tp
0
()sin(2 / )
∫
t
0
⎫
⎪
⎪
⎪
⎬
(2.1.8)
0
k
>
⎪
⎪
⎪
⎭
Coefficients
are related to
,
ab
kk
=>
ac k
,0
00
by
c
k
acc
=+
kk k
=+
bcc
kk k
−
−
⎫
⎪
⎬
⎪
⎭
. (2.1.9)
Again, as it was in obtaining (2.1.5), we can combine (2.1.7) and (2.1.8), that yields:
tp
+
0
ft f d
1
∼
() ( )
p
⎛⎞
∞
22()2()
⎜⎟
+τ + τ
∑
ppp
⎜⎟
k
=
⎝⎠
∫
t
0
tp
+
0
∫
1
t
0
ττ+
. (2.1.10)
⎛⎞
⎛⎞⎛⎞
πτ+ πτ+
() cos sin
⎜⎟
⎝⎠
ik t ik t
⎜⎟⎜⎟
⎝⎠⎝⎠
d
Remark 2.1.3
Generally speaking, Fourier series for any integrable function in an interval
,ab∈
[]
constructed with use of some other sets of the basic functions. These sets should satisfy
conditions of completeness, minimality, and orthogonality in the considered interval. Later on we
will discuss this problem in a more detail.
can be
2.1.2. The main properties of Fourier series
Proposition 2.1.1 (Dirichlet)
If function ()
finite number of points of discontinuity, then the Fourier series converges to
point, where
all points of discontinuity. The convergence in the points of continuity of function
called the convergence in the topology of simple convergence.
t is (i) bounded, (ii) periodic, (iii) having a finite number of extremums, and (iv)
is continuous, and converges to the average between left and right limits at
()ft
in every
()ft
is
()ft
Chapter 2. Fourier series, wavelets, and integral transforms
60
_____________________________________________________________________________
Remark 2.1.4
Conditions (i) – (iv) of the preceding theorem are known as the Dirichlet conditions; see
manuscript by Boyce and DiPrima for discussions.
Proposition 2.1.2 (Dirichlet and Riemann)
A. The Euler-Fourier coefficients of any periodic function satisfying Dirichlet conditions
satisfy the following asymptotic estimate:
1
cOk k
k
−
(),
=→∞
. (2.1.11)
If periodic function ( )f t poss
derivative satisfies the Dirichlet conditions, then the Euler-Fourier coefficients satisfy the
following asymptotic estimate:
cOk k
(),
=→∞. (2.1.12)
k
Proposition 2.1.3 (Presumably, Fourier)
Fourier series (2.1.3) or (2.1.7) converges in the
to any function integrable with its square in a closed interval
Proposition 2.1.4 (Parseval’s theorem)
If
()f t
and
are two functions defined in a closed interval
()g t
conditions, and expanded into the corresponding Fourier series
∞
() exp(2 / )
f taiktp
∼
∑
k
() exp(2 / )
gt b ikt p
=−∞
∞
∼
∑
k
=−∞
esses 1n − derivatives, that are continuous, and its n -th
(1)
n
−+
2
-topology (it is called converges on average)
L
,ab∈
[]
k
π
.
,ab∈
[]
, satisfying Dirichlet
(2.1.13)
k
π
Proof
then
b
() ()
f tgtdt ab
∫
a
∞
=
∑
=−∞
k
. (2.1.14)
kk
The proof follows from substituting the corresponding series (2.1.13) into integral in the lefthand side of (2.1.14) and remarking orthogonality of the exponents.
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