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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 2. Fourier series, wavelets, and integral transforms
x
f
fxy
fxyf
fxy
p
p
f
f
f
101
_____________________________________________________________________________
Remarks 2.5.3
A. Function
y
1
(, )
is called Poisson kernel associated with the upper half-plane. In view of (2.5.22)
Poisson integral transform can be written as convolution with Poisson kernel:
There is a generalization of the upper half-plane Poisson integral transform for functions
B.
defined in
integral transform is defined by
fxy f d x
(, ) ()
n
. Let
(1)
n
+
Γ
()
=η η∈
nn
(1)/2 (1)/2
++
π
Pxy
(, ) ( ) ,
xy f x P yd
=−ηηη
−∞
() ,1
fL pη∈ ≤ ≤∞
2
∫
n
−
=π
∞
∫
()
(
22
+
()
pn
()
y
2
−η +
xy
(2.5.22)
y
. (2.5.23)
and
2
y+∈
, then the corresponding
n
(2.5.24)
)
The Poisson integral (2.5.21) or (2.5.24) gives the solution of the Dirichlet problem for
C.
Laplace equation in the upper half-space and satisfying Dirichlet boundary condition at
.
0y →+
Proposition 2.5.5 (Convergence of Poisson transform for upper half-plane)
A.
Let
Function (, )
B.
() ,1
fL pη∈ ≤ <∞
then
almost everywhere in
at
.
0y →+
pn
()
lim ( , ) ( )
y
→+
n
;
belonging to
0
(, )
and
=
n
L at any fixed
()
be the corresponding Poisson transformation,
x
, (2.5.25)
0y >
converges to
Definition 2.5.5 (Unit disk Poisson integral transform)
The following integral transformation is called Poisson integral transform for a unit disk:
()fx
in
n
L
()
where
π
−
=π
() (2) ( )
z
is an integrable function on a unit disk and
1
∫
−π
it
ez
it
edt
+
it
−
ez
, (2.5.26)
.
1z <
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
102
_____________________________________________________________________________
Remarks 2.5.4
A. Integral in (2.5.26) can be expressed in terms of polar coordinates:
π
−
where
B.
There is generalization of the integral transform (2.5.27) on a unit ball in
()(dim) () , 0 1
() (2) ( )
=π
z
i
θ
zre
= .
′
=Σ σ σ ≤<
rx f d r
1
−π
−
1
∫
Σ
where Σ is the unit sphere in
Functions defined by (2.5.26) or (2.5.27) solve the Dirichlet problem for Laplace equation
C.
it
edt
∫
12cos
−θ−+
rtr
1
r
−
12cos
−γ+
rr
()
n
, γ is an angle between points σ∈Σ and x′∈Σ.
2
1
r
−
()
2
/2
n
2
(2.5.27)
2
n
:
, (2.5.28)
in the unit disk; analogously, function (2.5.28) solves the Dirichlet problem in a unit ball in
n
.
2.5.5. Weierstrass integral transform
Some of the basic properties of Weierstrass transform are discussed in Zayed (1996)
and Stein (1970).
Definition 2.5.6 (Weierstrass integral transform)
The following integral transformation is called Weierstrass integral transform:
∞
1/2
where
() 4 ()
ξ= π
is a (locally) integrable function; transformation parameter ξ can be complex.
()ft
−
()
fte dt
∫
−∞
Remarks 2.5.5
A. Since
∞
edt
−∞
2
t
−ξ−
/4
()
2
t
/4
−ξ−
()
1/2
4
=π∫ (2.5.30)
()
, (2.5.29)
Chapter 2. Fourier series, wavelets, and integral transforms
f
p
f
p
L
f
f
f
f
103
_____________________________________________________________________________
the factor
Weierstrass transformation.
A function
B.
1/24−
π
()
in (2.5.29) ensures constant functions in to be preserved under
1/2
() 4
Gt e
=π
−
()
2
t
/4
−
(2.5.31)
is known as Gaussian kernel. In view of (2.5.31) Weierstrass integral transform can
be written as convolution with Gaussian kernel:
∞
The Weierstrass transform is also called as Gauss or Gauss-Weierstrass transform
C.
() ( ) ()
ξ= ξ−η η η
fGd
∫
−∞
. (2.5.32)
due to its kernel being a Gaussian function.
Proposition 2.5.6 (Properties of Weierstrass integral transform)
A.
Let () ( ), 1
The Weierstrass transform is translation-invariant, meaning that the transform of the
B.
ft L p∈≤≤∞ and transform parameter ξ be real, then
p
function
Let ()
C.
p
f≥
(Weierstrass transform is bounded in
()
ta+ is
()
.
aξ+
t be a polynomial (not necessary homogeneous) of degree
polynomial of the same degree.
).
() ( )
fLξ∈
, then ()
n
p
ξ
is also a
and
Chapter 2. Fourier series, wavelets, and integral transforms
104
_____________________________________________________________________________
Bibliography to Chapter 2
ARENDT W., BATTY C.J.K., HIEBER M., and NEUBRANDER F.
Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2002. ISBN-13:
978-3764365493
B
HATIA R.
Fourier Series, Math. Assn. Amer., N.Y., 2004. ISBN-13: 978-0883857403
B
OCHNER S. and CHANDRASEKHARAN K.
Fourier Transforms, Annals of Mathematical Studies, Princeton University Press 1949, №19.
B
OYCE W.E. and DIPRIMA R.C.
Elementary Differential Equations and Boundary Value Problems, Eighth edition, John Wiley
& Sons, Inc., New Jersey, 2005. ISBN-13: 978-0470383346
B
RACEWELL R.N.
The Hartley Transform. N.Y., Oxford University Press, 1986. ISBN-13: 978-0195039696
B
RACEWELL R.N.
The Fourier Transform and Its Applications. 3rd ed. New York: McGraw-Hill, 1999. ISBN-13:
978-0073039381
B
URRUS C.S., GOPINATH R.A., and GUO H.
Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice Hall, New Jersey, 1998.
ISBN-13: 978-0134896007
C
HUI C.K.
An Introduction to Wavelets, Academic Press, San Diego, 1992. ISBN-13: 978-0121745844
C
OOLEY J.W. AND TUKEY J.W.
An algorithm for the machine calculation of complex Fourier series. Math. Comput. 1965,
pp. 297–301.
D
AVIES B.
Integral transforms and their applications, Third edition, Springer, N.Y., 2002.
D
EBNATH L. and BHATTA D.
Integral Transforms and their Applications. 2
1584885757
D
ELPRAT N., ESCUDIÉ B., GUILLEMAIN P., et al.
Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Trans.
Inf. Th., 1992,
38, pp. 644-664.
d
ed. Chapman & Hall, N.Y., 2007. ISBN-13: 978-
19,
YKE P.P.G.
D
An Introduction to Laplace Transforms and Fourier Series, Springer Verlag, London, 2001.
ISBN-13: 978-1852330156
F
LAJOLET P. and GOLIN M.
Chapter 2. Fourier series, wavelets, and integral transforms
105
_____________________________________________________________________________
Mellin transforms and asymptotics: The merge sort recurrence. Acta Informatica, 1994,
pp.673-696.
G
AUSS C.F.
Nachlass: Theoria interpolationis methodo nova tractata", 1866, Gauss Werke, Band 3,
Göttingen, Königliche Gesellschaft der Wissenschaften, pp. 265–327.
H
AAR A.
Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 1910,
371.
H
EIDEMAN M.T., JOHNSON D.H., and BURRUS C.S.
Gauss and the history of the fast Fourier transform. IEEE ASSP Magazine, 1984,
21.
J
ACKSON D.
Fourier Series and Orthogonal Polynomials, Dover Publ., N.Y., 2004. ISBN-13: 9780486438085
K
IERAT W. and SZTABA U.
Distribution, Integral Transforms and Applications, Taylor & Francis Inc, N.Y., 2003. ISBN13: 978-0415269582
69, pp 331-
1, (4), pp. 14–
31,
DE OLIVEIRA H.M. and ARAÚJO G.A.A.
Compactly Supported One-cyclic Wavelets Derived from Beta Distributions, Journal of
Communication and Information Systems, 2005,
ANDEY J.N.
P
The Hilbert Transform of Schwartz Distributions and Applications, Wiley-Interscience, N.Y.,
1995. ISBN-13: 978-0471033738
R
AMIREZ R.W.
The FFT: Fundamentals and Concepts. Englewood Cliffs, NJ: Prentice-Hall, 1984. ISBN-13:
978-0133143867
S
CHIFF J.L.
The Laplace Transform: Theory and Applications, Springer, N.Y., 1999.
S
CHWARTZ L.
Théorie des Distributions. I. 2
CHWARTZ L.
S
d
ed., Hermann, Paris, 1957.
Théorie des Distributions. II., Hermann, Paris, 1951.
S
CHWARTZ L.
Some Applications of the Theory of Distributions. Lectures on Modern Math. Vol.1, Springer,
N.Y., 1963.
S
TEIN E.M.
Singular Integrals and Differential Properties of Functions, Princeton Univ. Press, Princeton,
1971. ISBN-13: 978-0691080796
20, pp.27-33,
S
TEIN E.M. and WEISS G.
Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
ISBN-13: 978-0691080789
Chapter 2. Fourier series, wavelets, and integral transforms
106
_____________________________________________________________________________
T
ITCHMARSH E.C.
Introduction to the Theory of Fourier Integrals (3d edition), Clarendon Press, Oxford, 1986.
ISBN-13: 978-0828403245
T
ORRENCE C. and COMPO G.P.
A practical guide to wavelet analysis, Bulletin of the American Meteorological Society,
1998,vol.79, 61-78.
V
RETBLAD A.
Fourier Analysis and its Applications (Graduate Texts in Mathematics), Springer, N.Y., 2005.
ISBN-13: 978-1441918413
W
ALKER J.S.
Fast Fourier Transform, 2nd ed. Boca Raton, FL: CRC Press, 1996. ISBN-13: 978-0849371639
Z
AYED A.I.
Handbook of Functions and Generalized Function Transformation. CRC-Press, 1996. ISBN-13:
978-0849378515
Chapter 3.
107
Equation Chapter 3 Section 0
Theory of matrices
This chapter presents a formal and self-contained introduction to matrix algebra. It is
the purpose of this chapter to give an exhaustive presentation of the matrix algebra that
is mostly needed in various engineering applications.
The following monographs are possibly the best sources for general references: Collar
and Simpson (1987), Ciarlet (1989), Cullen (1990), Gantmaher (2005), Lutkepohl
(1996), Marcus and Minc (2008), and Meyer (2000).
The following monographs are mainly devoted to exposition of eigenvalues problems:
Barnett and Storey (1970), Chatelin (1993), and Wilkinson (1988).
Matrix norms and matrix inequalities are discussed by Beckenbach and Bellman
(1990), Belitskii and Lyubich (1988), Marcus and Mink (2010), see also Franklin
(1968).
Analyses of numerical methods in matrix algebra can be found in Anderson (2003),
Barnett (1990), Basilevsky (2005), Bjork et al. (1981), and Bronson (1989).
Tensors and properties of tensor invariants are discussed by Ericksen (1960).
3.1. Equation Chapter 3 Section 1 Elements
of matrix algebra
3.1.1. Basic definitions
Definition 3.1.1 (Matrix definition)
A matrix is a digital table of the form:
Chapter 3. Theory of matrices
M
108
__________________________________________________________________________
aaaa
⎛⎞
11 12 13 14
⎜⎟
aaaa
=
A
21 22 23 24
⎜⎟
⎜⎟
aaaa
31 32 33 34
⎝⎠
, (3.1.1)
where
34×
are some integer, real, or complex numbers. In (3.1.1) matrix A is a rectangular
a
ik
matrix. The following matrix is a square
bbb
⎛⎞
11 12 13
⎜⎟
bbb
=
B
21 22 23
⎜⎟
⎜⎟
bbb
31 32 33
⎝⎠
matrix:
33×
(3.1.2)
Identity (square) matrix is
100
⎛⎞
⎜⎟
=
I
010
⎜⎟
⎜⎟
001
⎝⎠
(3.1.3)
Sometimes, we will work with block matrices, whose elements are matrices of smaller
dimension:
AB
Block-matrix
to be well formed must have matrix components
C
number of the corresponding rows, while
⎛⎞
=
C
⎜⎟
DF
⎝⎠
(3.1.4)
,AB
and
with equal
,DF
,AD and ,BF must have equal number of the
corresponding columns.
All the matrices of the equal dimension
form a vector space denoted by
denoted by
.
n
M
,nm
. If
(n is number of rows, m is number of columns)
nm×
, then such a space of square matrices will be
nm=
3.1.2. Basic operations
Definition 3.1.2 (Matrices basic operations)
I.
The sum of matrices
with components
M∈C
,nm
II.
The product (convolution) of matrices
with components
of these matrices is denoted by
and B belonging to the same vector space
A
m
cab
=
∑
ij ik kj
k
1
=
, is a matrix
M
,nm
cab=+
ij ij ij
and
M∈A
,nm
M∈B
,mp
is a matrix
(this rule is known as “row by column”). The product
=⋅CAB
.
M∈C
,np
Chapter 3. Theory of matrices
M
M
109
__________________________________________________________________________
The full product (convolution) of two square matrices
III.
nn
cab
=⋅⋅=
AB
∑∑
11
ij
==
ij ji
.
,
is a scalar c :
∈AB
n
IV.
V.
Remark 3.1.1
A.
Let
B.
However, for complex vectors
The product of a number a by a matrix
components
cab=
ij ij
Trace (spur) of a square matrix
n
,
be two real vectors, a convolution operation
∈ab \
defines the scalar product in
. This is denoted by aB .
is a scalar defined by
∈A
n
n
(3.1.5)
∑
k
=
n
\ .
,
ab
1
∈ab ^
∑
k
kk
n
the scalar product is defined by
n
ab
k
k
1
=
⋅≡
ab
,
≡⋅≡
ab a b
(3.1.6)
is a matrix
M∈B
,nm
tr( )
A
M∈C
n
.
a
=
∑
kk
k
1
=
Thus, for complex vectors we will distinguish the scalar multiplication denoted by
and convolution. Expression (3.1.6) reveals that the scalar multiplication obeys
,⋅⋅
with
,nm
Hermitian condition:
,,=ab ba (3.1.7)
Proposition 3.1.1 (Properties of matrix operations)
I.
+=+AB BA
II.
III.
IV.
V.
VI.
++= ++ABC ABC
()()
⋅⋅= ⋅⋅ABC ABC
()()
+⋅=⋅+⋅ABCACBC
()
tr( ) tr( ) tr( )+= +AB A B (linearity of the trace)
tr( ) tr( )cc=AA
(additive commutativity)
(additive associativity)
(multiplicative associativity)
(distributivity)
(linearity of the trace)
Chapter 3. Theory of matrices
M
M
M
M
j
M
110
__________________________________________________________________________
Proposition 3.1.2 (Properties of matrix operations)
I.
⋅≠⋅AB BA (non-commutativity of multiplication)
II.
tr( ) tr( ) tr( )⋅≠AB A B (trace does not retain multiplicative structure)
Definition 3.1.3 (inverse matrix)
1
Inverse matrix for a given square matrix
condition:
∈A , is denoted by
n
−
∈A and satisfies
n
The inverse matrix exists
not for any square matrix. If the inverse matrix exists, then matrix A
⋅=⋅=AA A A I
is called non-degenerate or non-singular.
Proposition 3.1.3 (Property of the trace)
If
is a non-singular square matrix, then for any square matrix
∈W
n
tr tr
()
Definition 3.1.4 (Transposition of a matrix)
A transposed matrix
with components:
M
,mn
t
for the given matrix
A
Operation of transposition is well defined for
11−−
1
⋅⋅ =WAW A
−
t
aa= . (3.1.10)
()
ij
. (3.1.8)
:
∈A
n
(3.1.9)
()
is a matrix belonging to the space
M∈A
,nm
i
any matrix.
Definition 3.1.5 (Complex conjugate matrix for a given matrix)
A complex conjugate matrix *A for the given (complex) matrix
matrix:
t
*
. (3.1.11)
=AA
Thus, the complex-conjugate matrix for a given real matrix coincides with the transposed
matrix.
Example 3.1.1
Let
:
∈A
2
is the following
M∈A
,nm
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