-
logo.jpg logo.bb
. . . . . .
.
......Polynomial Wavelet Type Expansions on the Spline
A. Askari-Hemmat1, M. A. Dehghan2, M. A. Skopina3
1Department of Mathematics, Graduate University of Advanced
Technology, Kerman, Iran1Department of Mathematics, Shahid Bahonar
University, Kerman, Iran
2Department of Mathematics, Vali-E-Asr University, Rafsanjan,
Iran3Department of Mathematics, St. Petersburg University,
Russia
.. Jump to one slide .. Jump to second slide
Jan., 06, 2014Askari-Hemmat ([email protected]) Polynomial
Wavelet Type Expansions on the Spline Jan., 06, 2014 1 / 22
-
logo.jpg logo.bb
. . . . . .
Introduction
.
......
We present a polynomial wavelet-type system on Sd such that
anycontinuous function can be expanded with respect to these
wavelets.The order of the growth of the degree of the polynomials
is optimal.The coefficients in the expansion are the inner product
of the function andthe corresponding element of a dual wavelet
system.The dual wavelet system is also a polynomial system with the
same growthof degree of polynomials. The system is redundant.
.
......
A construction of a polynomial basis is also presented. In
constrast to ourwavelet-type system, this basis is not suitable for
implementation, becauseof two drawbacks: first there are no
explicit formoula for the coefficientfunctionals and, secend, the
growth of the degree of polynomials is toorapid.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 2 / 22
-
logo.jpg logo.bb
. . . . . .
Introduction
.
......
We present a polynomial wavelet-type system on Sd such that
anycontinuous function can be expanded with respect to these
wavelets.The order of the growth of the degree of the polynomials
is optimal.The coefficients in the expansion are the inner product
of the function andthe corresponding element of a dual wavelet
system.The dual wavelet system is also a polynomial system with the
same growthof degree of polynomials. The system is redundant.
.
......
A construction of a polynomial basis is also presented. In
constrast to ourwavelet-type system, this basis is not suitable for
implementation, becauseof two drawbacks: first there are no
explicit formoula for the coefficientfunctionals and, secend, the
growth of the degree of polynomials is toorapid.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 2 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
History
.
......
1961, Foias and Singer found the first polynomial basis for C[a,
b].
1987, Privalov, consrucrtd optimal polynomial basis for the
space ofcontinuous functions in both the trigonometric and the
algebraiccases.
1994, Lorentz and Saakyan , proved that packets of periodic
Meyerwavelets are such bases. That is orthogonal.
1988, Freeden and Schreiner, proposed the wavelet-type
polynomialsystems on the two-dimensional sphere.
1988, Farkov extended their construction to the
multidimensionalcase.
2001, Skopina, found a similar wavelet system in L2[a, b] and
provedthat the corresponding wavelet packets are optimal
plynomialorthogonal bases for the space C[a, b].
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 3 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
We show that, for any function f ∈ C(Sd), certain spherical
waveletsprovide a uniformly convergent polynomial expansion.
.
......
The following notations will be used:
The inner product
x.y = x1y1 + · · ·+ xdyd.
πdn : is the space of polynomials in d variables of degree at
most n.
Bd ={x ∈ Rd|∥x∥ ≤ 1
}, Sd = ∂Bd+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 4 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
We show that, for any function f ∈ C(Sd), certain spherical
waveletsprovide a uniformly convergent polynomial expansion.
.
......
The following notations will be used:
The inner product
x.y = x1y1 + · · ·+ xdyd.
πdn : is the space of polynomials in d variables of degree at
most n.
Bd ={x ∈ Rd|∥x∥ ≤ 1
}, Sd = ∂Bd+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 4 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
We show that, for any function f ∈ C(Sd), certain spherical
waveletsprovide a uniformly convergent polynomial expansion.
.
......
The following notations will be used:
The inner product
x.y = x1y1 + · · ·+ xdyd.
πdn : is the space of polynomials in d variables of degree at
most n.
Bd ={x ∈ Rd|∥x∥ ≤ 1
}, Sd = ∂Bd+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 4 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
We show that, for any function f ∈ C(Sd), certain spherical
waveletsprovide a uniformly convergent polynomial expansion.
.
......
The following notations will be used:
The inner product
x.y = x1y1 + · · ·+ xdyd.
πdn : is the space of polynomials in d variables of degree at
most n.
Bd ={x ∈ Rd|∥x∥ ≤ 1
}, Sd = ∂Bd+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 4 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
We show that, for any function f ∈ C(Sd), certain spherical
waveletsprovide a uniformly convergent polynomial expansion.
.
......
The following notations will be used:
The inner product
x.y = x1y1 + · · ·+ xdyd.
πdn : is the space of polynomials in d variables of degree at
most n.
Bd ={x ∈ Rd|∥x∥ ≤ 1
}, Sd = ∂Bd+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 4 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
G(λ)n : Denotes the nth Gegenbauer polynomial of order λ.
Jacobi’s polynomials, {P (α,β)n (x)}, are defined as
orthogonalpolynomials with respect to the weight function (1− x)α(1
+ x)β on(−1, 1). (α > −1, β > −1)If we put α = β in the above
definition and define A
(λ)0 = 1,
A(0)n =2n+1(n− 1)
1.3. . . . .(2n− 1), n ≥ 1
A(λ)n =2n(2λ)(2λ+ 1) · · · (2λ+ n− 1)
(2λ+ 1) · · · (2(λ+ n)− 1), n ≥ 1, λ ̸= 0
then Gegenbauer polynomials are defined as
G(λ)n = A(λ)n P
(α,α)n (x), λ = (α+
1
2), n ≥ 0.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 5 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
G(λ)n : Denotes the nth Gegenbauer polynomial of order λ.
Jacobi’s polynomials, {P (α,β)n (x)}, are defined as
orthogonalpolynomials with respect to the weight function (1− x)α(1
+ x)β on(−1, 1). (α > −1, β > −1)If we put α = β in the above
definition and define A
(λ)0 = 1,
A(0)n =2n+1(n− 1)
1.3. . . . .(2n− 1), n ≥ 1
A(λ)n =2n(2λ)(2λ+ 1) · · · (2λ+ n− 1)
(2λ+ 1) · · · (2(λ+ n)− 1), n ≥ 1, λ ̸= 0
then Gegenbauer polynomials are defined as
G(λ)n = A(λ)n P
(α,α)n (x), λ = (α+
1
2), n ≥ 0.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 5 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
The area of d-dimensional sphere (Sd):
ωd =
∫sdds(x) =
2π(d+12
)
Γ(d+12 ).
For functions F,G ∈ L2(Sd), the inner product is defined as
follows.
< F,G >=
∫Sd
F (x)Ḡ(x)ds(x).
The restriction to Sd of a homogeneous harmonic polynomial
ofdegree n is called a spherical harmonic of degree n.
For a fixed integer n, Hdn denotes the class of spherical
harmonics ofdegree n.
The spaces Hdn are mutually orthogonal.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 6 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
The area of d-dimensional sphere (Sd):
ωd =
∫sdds(x) =
2π(d+12
)
Γ(d+12 ).
For functions F,G ∈ L2(Sd), the inner product is defined as
follows.
< F,G >=
∫Sd
F (x)Ḡ(x)ds(x).
The restriction to Sd of a homogeneous harmonic polynomial
ofdegree n is called a spherical harmonic of degree n.
For a fixed integer n, Hdn denotes the class of spherical
harmonics ofdegree n.
The spaces Hdn are mutually orthogonal.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 6 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
The area of d-dimensional sphere (Sd):
ωd =
∫sdds(x) =
2π(d+12
)
Γ(d+12 ).
For functions F,G ∈ L2(Sd), the inner product is defined as
follows.
< F,G >=
∫Sd
F (x)Ḡ(x)ds(x).
The restriction to Sd of a homogeneous harmonic polynomial
ofdegree n is called a spherical harmonic of degree n.
For a fixed integer n, Hdn denotes the class of spherical
harmonics ofdegree n.
The spaces Hdn are mutually orthogonal.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 6 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
The area of d-dimensional sphere (Sd):
ωd =
∫sdds(x) =
2π(d+12
)
Γ(d+12 ).
For functions F,G ∈ L2(Sd), the inner product is defined as
follows.
< F,G >=
∫Sd
F (x)Ḡ(x)ds(x).
The restriction to Sd of a homogeneous harmonic polynomial
ofdegree n is called a spherical harmonic of degree n.
For a fixed integer n, Hdn denotes the class of spherical
harmonics ofdegree n.
The spaces Hdn are mutually orthogonal.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 6 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
The area of d-dimensional sphere (Sd):
ωd =
∫sdds(x) =
2π(d+12
)
Γ(d+12 ).
For functions F,G ∈ L2(Sd), the inner product is defined as
follows.
< F,G >=
∫Sd
F (x)Ḡ(x)ds(x).
The restriction to Sd of a homogeneous harmonic polynomial
ofdegree n is called a spherical harmonic of degree n.
For a fixed integer n, Hdn denotes the class of spherical
harmonics ofdegree n.
The spaces Hdn are mutually orthogonal.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 6 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
⊕nl=1H
dl comprises the restrictions to of all algebraic polynomials
in
d+ 1 variables of degree not exceeding n.
Dimension of Hdn is given by:
δdn := dimHdn =
{2n+d+1n+d−1
(n+d−1
n
)if n ≥ 1
1 if n = 0
⊕∞n=1H
dn is dense in L2(S
d).
Muller’s formula (1966): Let{Ynk, k = 1, · · · δdn be an
orthonormal
basis for Hdn, then
δdn∑k=0
Ynk(x)Ynk(y) =(2n+ d− 1)ωd(d− 1)
Gd−12
n (x.y) (1)
for all x, y ∈ Sd and for all n = 0, 1, . . . .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 7 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
⊕nl=1H
dl comprises the restrictions to of all algebraic polynomials
in
d+ 1 variables of degree not exceeding n.
Dimension of Hdn is given by:
δdn := dimHdn =
{2n+d+1n+d−1
(n+d−1
n
)if n ≥ 1
1 if n = 0
⊕∞n=1H
dn is dense in L2(S
d).
Muller’s formula (1966): Let{Ynk, k = 1, · · · δdn be an
orthonormal
basis for Hdn, then
δdn∑k=0
Ynk(x)Ynk(y) =(2n+ d− 1)ωd(d− 1)
Gd−12
n (x.y) (1)
for all x, y ∈ Sd and for all n = 0, 1, . . . .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 7 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
⊕nl=1H
dl comprises the restrictions to of all algebraic polynomials
in
d+ 1 variables of degree not exceeding n.
Dimension of Hdn is given by:
δdn := dimHdn =
{2n+d+1n+d−1
(n+d−1
n
)if n ≥ 1
1 if n = 0
⊕∞n=1H
dn is dense in L2(S
d).
Muller’s formula (1966): Let{Ynk, k = 1, · · · δdn be an
orthonormal
basis for Hdn, then
δdn∑k=0
Ynk(x)Ynk(y) =(2n+ d− 1)ωd(d− 1)
Gd−12
n (x.y) (1)
for all x, y ∈ Sd and for all n = 0, 1, . . . .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 7 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
⊕nl=1H
dl comprises the restrictions to of all algebraic polynomials
in
d+ 1 variables of degree not exceeding n.
Dimension of Hdn is given by:
δdn := dimHdn =
{2n+d+1n+d−1
(n+d−1
n
)if n ≥ 1
1 if n = 0
⊕∞n=1H
dn is dense in L2(S
d).
Muller’s formula (1966): Let{Ynk, k = 1, · · · δdn be an
orthonormal
basis for Hdn, then
δdn∑k=0
Ynk(x)Ynk(y) =(2n+ d− 1)ωd(d− 1)
Gd−12
n (x.y) (1)
for all x, y ∈ Sd and for all n = 0, 1, . . . .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 7 / 22
-
logo.jpg logo.bb
. . . . . .
.Theorem 1: F. Mahaskar and others (2001)..
......
There is constants Nd and Ad depending only on d so that for any
finiteset {ηl}l∈Ω of distinct points ηl ∈ Sd and for any positive
integer N ≥ Ndsatisfying
N maxx∈Sd
minl∈Ω
|x− ηl| ≤ Ad
there exist nonnegative weights al, l ∈ Ω such that∫Sd
P (x)ds(x) =∑l∈Ω
alP (ηl)
for all P ∈ πd+1N .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 8 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Because of this theorem, to each positive integer j we can
assign a set
{η(j)l }l∈Ωj of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights the following
properties:card(Ωj)∼ 2dj , ∫
Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l )
for any P ∈ πd+12j
. Additionally, we introduce the set Ω0 := {0} and puta(0)0 =
ωd.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 9 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Due to this theorem to each positive integer j we can assign a
set
{η(j)l } of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights with the
following properties:
card(Ωj)∼ 2dj . ∫Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l ) (2)
for any P ∈ πd+12j
.
Additionally we introduce the set Ω0 := {0} and put a(0)0 =
ωd.
Let hj(n) =(2
j−n+d−1d )
(2j+d−12j−1 )
for n = 0, ..., 2j − 1, hj(n) = 0 for
n = 2j , 2j + 1, ....
An inequality due to kogbeliantz states that
n∑n=0
(N − n+ d
d
)(2n+d−1)G(
d−12
)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 10 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Due to this theorem to each positive integer j we can assign a
set
{η(j)l } of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights with the
following properties:
card(Ωj)∼ 2dj .
∫Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l ) (2)
for any P ∈ πd+12j
.
Additionally we introduce the set Ω0 := {0} and put a(0)0 =
ωd.
Let hj(n) =(2
j−n+d−1d )
(2j+d−12j−1 )
for n = 0, ..., 2j − 1, hj(n) = 0 for
n = 2j , 2j + 1, ....
An inequality due to kogbeliantz states that
n∑n=0
(N − n+ d
d
)(2n+d−1)G(
d−12
)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 10 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Due to this theorem to each positive integer j we can assign a
set
{η(j)l } of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights with the
following properties:
card(Ωj)∼ 2dj . ∫Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l ) (2)
for any P ∈ πd+12j
.
Additionally we introduce the set Ω0 := {0} and put a(0)0 =
ωd.
Let hj(n) =(2
j−n+d−1d )
(2j+d−12j−1 )
for n = 0, ..., 2j − 1, hj(n) = 0 for
n = 2j , 2j + 1, ....
An inequality due to kogbeliantz states that
n∑n=0
(N − n+ d
d
)(2n+d−1)G(
d−12
)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 10 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Due to this theorem to each positive integer j we can assign a
set
{η(j)l } of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights with the
following properties:
card(Ωj)∼ 2dj . ∫Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l ) (2)
for any P ∈ πd+12j
.
Additionally we introduce the set Ω0 := {0} and put a(0)0 =
ωd.
Let hj(n) =(2
j−n+d−1d )
(2j+d−12j−1 )
for n = 0, ..., 2j − 1, hj(n) = 0 for
n = 2j , 2j + 1, ....
An inequality due to kogbeliantz states that
n∑n=0
(N − n+ d
d
)(2n+d−1)G(
d−12
)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 10 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Due to this theorem to each positive integer j we can assign a
set
{η(j)l } of distinct points η(j)l ∈ S
d and a set {a(j)l }l∈Ωj of nonnegativeweights with the
following properties:
card(Ωj)∼ 2dj . ∫Sd
P (x)ds(x) =∑l∈Ωj
a(j)l P (η
(j)l ) (2)
for any P ∈ πd+12j
.
Additionally we introduce the set Ω0 := {0} and put a(0)0 =
ωd.
Let hj(n) =(2
j−n+d−1d )
(2j+d−12j−1 )
for n = 0, ..., 2j − 1, hj(n) = 0 for
n = 2j , 2j + 1, ....
An inequality due to kogbeliantz states that
n∑n=0
(N − n+ d
d
)(2n+d−1)G(
d−12
)n (t) ≥ 0, for all t ∈ [−1, 1]. (3)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 10 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Now we are going to define our wavelet-type functions.Set for j
= 1, ..., n = 0, 1, ...:
gj(n) = hj(n) + hj−1(n)
g̃j(n) = hj(n)− hj−1(n)
g0(0) = h0(0) + 1, g̃0(0) = h0(0)− 1, g0(n) = g̃0(n) = 0, for n
= 1, 2, ....
For each nonnegative integer j and for each l ∈ Ωj+1, define the
waveletfunction Ψjl, the dual wavelet function Ψ̃jl and the scaling
functionΦ(j+1)l by
Ψjl(x) =1
ωd(d− 1)∑n∈z+
gj(n)(2n+ d− 1)×G( d−1
2)
n (η(j+1)l . x)
Ψ̃jl(x) =1
ωd(d− 1)∑n∈z+
g̃j(n)(2n+ d− 1)×G( d−1
2)
n (η(j+1)l . x)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 11 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Now we are going to define our wavelet-type functions.Set for j
= 1, ..., n = 0, 1, ...:
gj(n) = hj(n) + hj−1(n)
g̃j(n) = hj(n)− hj−1(n)
g0(0) = h0(0) + 1, g̃0(0) = h0(0)− 1, g0(n) = g̃0(n) = 0, for n
= 1, 2, ....For each nonnegative integer j and for each l ∈ Ωj+1,
define the waveletfunction Ψjl, the dual wavelet function Ψ̃jl and
the scaling functionΦ(j+1)l by
Ψjl(x) =1
ωd(d− 1)∑n∈z+
gj(n)(2n+ d− 1)×G( d−1
2)
n (η(j+1)l . x)
Ψ̃jl(x) =1
ωd(d− 1)∑n∈z+
g̃j(n)(2n+ d− 1)×G( d−1
2)
n (η(j+1)l . x)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 11 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Φ(j+1)l(x) =1
ωd(d− 1)∑n∈z+
h(n)(2n+ d− 1)×G(d−12
)n (η
(j+1)l . x)
Complete this collection by the function Φ0,0 =1ωd
and set Φ0 =√ωdΦ0,0.
For F ∈ C(Sd), we will study the convergence of the series
< F,Φ0 > Φ0 +∞∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil.
Set
Λj,ω(F ) =< F,Φ0 > Φ0+
j−1∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil
+∑l∈ω
a(j+1)l < F, Ψ̃jl > Ψjl
where ω is a subset of Ωj+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 12 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Φ(j+1)l(x) =1
ωd(d− 1)∑n∈z+
h(n)(2n+ d− 1)×G(d−12
)n (η
(j+1)l . x)
Complete this collection by the function Φ0,0 =1ωd
and set Φ0 =√ωdΦ0,0.
For F ∈ C(Sd), we will study the convergence of the series
< F,Φ0 > Φ0 +∞∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil.
Set
Λj,ω(F ) =< F,Φ0 > Φ0+
j−1∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil
+∑l∈ω
a(j+1)l < F, Ψ̃jl > Ψjl
where ω is a subset of Ωj+1.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 12 / 22
-
logo.jpg logo.bb
. . . . . .
.
......
Φ(j+1)l(x) =1
ωd(d− 1)∑n∈z+
h(n)(2n+ d− 1)×G(d−12
)n (η
(j+1)l . x)
Complete this collection by the function Φ0,0 =1ωd
and set Φ0 =√ωdΦ0,0.
For F ∈ C(Sd), we will study the convergence of the series
< F,Φ0 > Φ0 +∞∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil.
Set
Λj,ω(F ) =< F,Φ0 > Φ0+
j−1∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil
+∑l∈ω
a(j+1)l < F, Ψ̃jl > Ψjl
where ω is a subset of Ωj+1.Askari-Hemmat ([email protected])
Polynomial Wavelet Type Expansions on the Spline Jan., 06, 2014 12
/ 22
-
logo.jpg logo.bb
. . . . . .
.Main Theorem..
......
For any F ∈ C(Sd)limj→∞
∥F − Λj,ω(F )∥ = 0 (4)
First we will prove that the operators Λj,ω : C(Sd) −→ C(Sd),
are
uniformly bounded. We show that (4) holds on the set of
sphericalpolynomials and our claim is a consequence of
Banach-Steinhaustheorem. To prove the main theorem we need the
following lemma:
.Lemma..
......
For any F ∈ C(Sd)
< F,Φ0 > Φ0+
j−1∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil
=∑l∈Ωj
a(j)l < F,Φjl > Φjl. (5)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 13 / 22
-
logo.jpg logo.bb
. . . . . .
.Main Theorem..
......
For any F ∈ C(Sd)limj→∞
∥F − Λj,ω(F )∥ = 0 (4)
First we will prove that the operators Λj,ω : C(Sd) −→ C(Sd),
are
uniformly bounded. We show that (4) holds on the set of
sphericalpolynomials and our claim is a consequence of
Banach-Steinhaustheorem. To prove the main theorem we need the
following lemma:
.Lemma..
......
For any F ∈ C(Sd)
< F,Φ0 > Φ0+
j−1∑i=0
∑l∈Ωi+1
a(i+1)l < F, Ψ̃il > Ψil
=∑l∈Ωj
a(j)l < F,Φjl > Φjl. (5)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 13 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.
......
The following theorem due to Krein-Milman-Rutman plays the
mainrole:Let {Qk}∞k=0 be a basis for a Banach space H and let αk ∈
H∗,k = 0, 1, ..., be coefficient functionals for this basis. If Pk
∈ Hand ∥Pk −Qk∥ ≤ 2
−k−2
∥αk∥ =: λk for all k = 0, 1, ..., then the
sequence {Pk}∞k=0 is a basis for H.
If we can find a basis, say {Qn}, for C(Sd) and set λk =
2−k−2
∥αk∥ where
{αk}∞k=1 is the sequence of corresponding coefficient
functional.So to construct a polynomial basis for C(Sd) it’s enough
to constructa basis for it.
We start with finding an initial basis {Qk} for C(Sd)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 14 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.
......
The following theorem due to Krein-Milman-Rutman plays the
mainrole:Let {Qk}∞k=0 be a basis for a Banach space H and let αk ∈
H∗,k = 0, 1, ..., be coefficient functionals for this basis. If Pk
∈ Hand ∥Pk −Qk∥ ≤ 2
−k−2
∥αk∥ =: λk for all k = 0, 1, ..., then the
sequence {Pk}∞k=0 is a basis for H.If we can find a basis, say
{Qn}, for C(Sd) and set λk = 2
−k−2
∥αk∥ where
{αk}∞k=1 is the sequence of corresponding coefficient
functional.So to construct a polynomial basis for C(Sd) it’s enough
to constructa basis for it.
We start with finding an initial basis {Qk} for C(Sd)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 14 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.
......
The following theorem due to Krein-Milman-Rutman plays the
mainrole:Let {Qk}∞k=0 be a basis for a Banach space H and let αk ∈
H∗,k = 0, 1, ..., be coefficient functionals for this basis. If Pk
∈ Hand ∥Pk −Qk∥ ≤ 2
−k−2
∥αk∥ =: λk for all k = 0, 1, ..., then the
sequence {Pk}∞k=0 is a basis for H.If we can find a basis, say
{Qn}, for C(Sd) and set λk = 2
−k−2
∥αk∥ where
{αk}∞k=1 is the sequence of corresponding coefficient
functional.So to construct a polynomial basis for C(Sd) it’s enough
to constructa basis for it.
We start with finding an initial basis {Qk} for C(Sd)
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 14 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 1...
......
Counstruction a basis for the space C([0, 1]d) and
C0([0, 1]d) :=
{f ∈ C([0, 1]d) : f(x) = 0, ∀x ∈ ∂([0, 1]d)
}.
Let {fn}∞n=0 be the Faber-Cshaude basis for C[0, 1] defined
by
f0(x) = 1, x ∈ [0, 1]
f1(x) = x, x ∈ [0, 1],
for n = 2k + i, k = 0, 1, ..., i = 1, 2, ..., 2k, fn is linear
and continuous on[ i−12k
, 2i−12k+1 ] and on [2i−12k+1
, i2k],
fn(x) =
{0 if x /∈ ( i−1
2k, i2k)
1 if x = 2i−12k+1
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 15 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 1...
......
The tensor product of d systems fn is a basis for C([0, 1]d),
say
B = {Fk}∞k=1. Elements of B are functions of the form
F (x) = fn1(x)fn2(x) · · · fnd(x), xj ∈ [0, 1], nj ∈ Z+, j = 1,
· · · , d.
Denote by B′ the subset of B that contains of all the functions
F suchthat nj ̸= 0, nj ̸= 1, j = 1, · · · , d. Then B′ is a basis
foe C0([0, 1]d).
.Step 2...
......
Construction of basis for the space C(Bd) and
C0(Bd) :=
{f ∈ C(Bd) : f(x) = 0, ∀x ∈ ∂Bd
}.
By a change of variable, we can replace [0, 1]d by [−1, 1]d,
preserving thesame notation B, B′ for the corresponding basis.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 16 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 1...
......
The tensor product of d systems fn is a basis for C([0, 1]d),
say
B = {Fk}∞k=1. Elements of B are functions of the form
F (x) = fn1(x)fn2(x) · · · fnd(x), xj ∈ [0, 1], nj ∈ Z+, j = 1,
· · · , d.
Denote by B′ the subset of B that contains of all the functions
F suchthat nj ̸= 0, nj ̸= 1, j = 1, · · · , d. Then B′ is a basis
foe C0([0, 1]d).
.Step 2...
......
Construction of basis for the space C(Bd) and
C0(Bd) :=
{f ∈ C(Bd) : f(x) = 0, ∀x ∈ ∂Bd
}.
By a change of variable, we can replace [0, 1]d by [−1, 1]d,
preserving thesame notation B, B′ for the corresponding
basis.Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 16 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 2...
......
Define a mapϕ : Bd −→ [−1, 1]d
ϕ(x) = (ρr(θ), θ)
where x = (ρ, θ), 0 ≤ ρ ≤ 1, θ ∈ Sd−1 and r(θ) is the length of
thesegment {
y = (t, θ) : t ≥ 0, y ∈ [−1, 1]d}.
Then θ is bijective and the functions Gk := Fk(ϕ), k = 1, 2,
..., constitutea basis for C(Bd). Denote this basis by B′.A basis
B′′ = {G
(0)k }
∞k=1 for C0(B
d) can be constructed similarly fromB′.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 17 / 22
-
logo.jpg logo.bb
. . . . . .
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 18 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 3...
......
Construction of a basis for C(Sd).
Let C(o)(Sd) and C(e)(Sd) be respectively the set of functions f
∈ C(Sd)
such that
f(x1, x2, ..., xd, xd+1) = f(x1, x2, ..., xd,−xd+1)
and the set of functions f ∈ C(Sd) such that
f(x1, x2, ..., xd, xd+1) = −f(x1, x2, ..., xd,−xd+1).
Each f ∈ C(Sd) can be represented in the form f = f (e) + f (o),
wheref (e) ∈ C(e)(Sd), f (o) ∈ C(o)(Sd). It is obvious that if
{H(e)k }
∞k=1 and
{H(o)k }∞k=1 are bases for C
(e)(Sd) and C(o)(Sd), then the system
{H(e)k ,H(o)k }
∞k=1 is a basis for C(S
d).
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 19 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 3...
......
Construction of a basis for C(Sd).
Let C(o)(Sd) and C(e)(Sd) be respectively the set of functions f
∈ C(Sd)
such that
f(x1, x2, ..., xd, xd+1) = f(x1, x2, ..., xd,−xd+1)
and the set of functions f ∈ C(Sd) such that
f(x1, x2, ..., xd, xd+1) = −f(x1, x2, ..., xd,−xd+1).
Each f ∈ C(Sd) can be represented in the form f = f (e) + f (o),
wheref (e) ∈ C(e)(Sd), f (o) ∈ C(o)(Sd). It is obvious that if
{H(e)k }
∞k=1 and
{H(o)k }∞k=1 are bases for C
(e)(Sd) and C(o)(Sd), then the system
{H(e)k ,H(o)k }
∞k=1 is a basis for C(S
d).
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 19 / 22
-
logo.jpg logo.bb
. . . . . .
Construct a polynomial basis for C(Sd).
.Step 3...
......
So it remaines to find bases for C(e)(Sd) and C(o)(Sd).
For x = (x1, x2, ..., xd+1) ∈ Sd, set
H(e)k (x) = Gk(x1, x2, ..., xd)
H(o)k (x) =
{G
(o)k (x1, x2, ..., xd) if xd+1 ≥ 0
−G(o)k (x1, x2, ..., xd) if xd+1 ≤ 0
{H(e)k }∞k=1 and {H
(o)k }
∞k=1 are the required bases.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 20 / 22
-
logo.jpg logo.bb
. . . . . .
REFERENCES
.
......
1. C. Foias and I. Singer, Some remarks on strongly independent
sequences and bases inBanach spaces, Rev. Math. Pure et Appl. Acad.
R.P.R., 6 (1961), no. 3, 589–594.2. Al. A. Privalov, On the growth
of degrees of polynomial bases and approximation oftrigonometric
projections, Mat. Zametki [Math Notes], 42 (1987) no. 2, 207–214.3.
R. A. Lorentz and A. A. Saakyan (Sahakian), Orthogonal
trigonometric Schauderbases of optimal degree for C(0, 2π), J.
Fourier Anal. Appl., 1 (1994), no. 1, 103–112.4. M. A. Skopina,
Orthogonal polynomial Schauder bases for C[−1, 1] of optimaldegree,
Mat. Sb. [Russian Acad. Sci. Sb. Math.], 192 (2001), no. 3,
115–136.5. W. Freeden and M. Schreiner, Orthogonal and
non-orthogonal multiresolutionanalysis, scale discrete and exact
fully discrete wawelet transform on the sphere,Constructive
Approximation, 14 (1998), 493–515.6. Yu. Farkov, B-spline wavelets
on the sphere, in: Self-Similar Systems, Proceedings ofthe
International Workshop (July 30–August 7, 1998, Dubna, Russia),
JINR, E5-99-38,Dubna, 1999, pp. 79–82.7. M. Skopina, Polynomial
expansions of continuous functions on the sphere and on thedisk,
in: Preprint no. 2001:5, University of South Carolina, Department
of Mathematics,2001.8. C. Müller, Spherical Harmonics, Lecture
Notes in Math., vol. 17, Springer-Verlag,Berlin, 1966.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 21 / 22
-
logo.jpg logo.bb
. . . . . .
REFERENCES
.
......
9. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on
Euclidean Spaces,Princeton Univ. Press, Princeton, NJ, 1971.10. H.
N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical
Marcinkiewicz–Zygmundinequalities and positive quadrature, Math.
Comp., 70 (2001), no. 235, 1113–1130.11. H. N. Mhaskar, F. J.
Narcowich, J. D. Ward, and J. Prestin, Polynomial frames onthe
sphere, Adv. Comput. Math., 13 (2000), 387–403.12. R. Askary,
Orthogonal Polynomials and Spherical Functions, SIAM,
Phildelphia,1975.13. G. Szegö, Orthogonal Polynomials, Amer. Math.
Soc., New York, 1959.14. B. S. Kashin and A. A. Saakyan, Orthogonal
Series, [in Russian], Izd, AFTs,Moscow, 1999.
15. Wang Kunyang and Li Luoking, Harmonic Analysis and
Approximation on the Unit
Sphere, Graduate Series in Mathematics, 2000.
Askari-Hemmat ([email protected]) Polynomial Wavelet Type
Expansions on the Spline Jan., 06, 2014 22 / 22