World Applied Sciences Journal 22 (Special Issue of Applied Math): 122-129, 2013 ISSN 1818-4952 c
IDOSI Publications, 2013 DOI:10.5829/idosi.wasj.22.am.12.2013
Tensorial approximate identities for vector valued functions M. Akram Department of Mathematics, GC University, Katchery Road Lahore, 54000, Pakistan
[email protected],
[email protected] Abstract: In this article, we introduce a tensorial kernel for vector valued functions which is compactly supported on a bounded region V in R3 . It is proved that the convolutions of the derived kernels with a vector valued function f having continuous first derivatives converges to f , i.e. the defining property of an approximate identity, is proved. Further, some numerical tests are shown. Key words: Green’s function, weak convergence, approximate identity, tensorial kernel.
Let V be an open and bounded subset of R3 , we denote the set of functions F : V → R with continuous derivatives of all orders by C∞ (V ), the set of functions f : V → R3 with continuous derivatives of all orders by c∞ (V ) and and for every compact subset K of V , ∞ C∞ K (V ) is the set of all functions in C (V ) with sup∞ port in K. CK (V ) is a linear space and the topology which makes it into a Fréchet space is induced by the seminorms
INTRODUCTION The rule of an approximate identity is as follows: A function f which has to be approximated is convolved with a kernel Kδ for some δ. If the kernel satisfies certain conditions then the convolutions converge in a certain limit such as δ → 0+ to f . Note that approximate identities for scalar function on balls in R3 are studied e.g. in [14]. Further works on localizing kernels, like scaling functions and wavelets, on the 3D ball can e.g. be found in [3, 5, 8, 12, 13, 15]. In this paper we show how tensorial kernels on a 3-dimensional bounded region can be constructed. We prove that these kernels establish an approximate identity for all vector valued functions with continuous first derivative. The convergence established in this case is in the sense of a distribution. Particular practical relevance is the case when the considered bounded region is a ball. An example of an application of approximating structures on a three-dimensional bounded region can be found in geophysics. There, the choice of appropriate tools for describing features of the Earth’s interior, such as the mass density, the speed of propagation of seismic P and S waves and other rheological quantities, is still a field of research. Moreover, the approximation of vectorial fields such as currents on a ball is also relevant in medical imaging.
Pm (φ) = sup {|∂ α φ(x)| : x ∈ K, |α| ≤ m} , φ ∈ C∞ K (V ), m ∈ N0 , with the sets Dm (r) = {φ ∈ C∞ K (V ) : Pm (φ) < r} , as a local base (see [4]). Where, a local base at a point x, we means, a collection of base open sets in which every open set containing x contains one of these base sets, which contains x. Moreover, C∞ K (V ) is also a closed ∞ subspace of C0 (V ), where we define the topology of ∞ C∞ 0 (V ) := ∪K⊂V CK (V )
to be the finest locally convex topology for which every linear functional defined on C∞ 0 (V ) is continuous if and only if its restriction to C∞ (V ) is continuous for every K K ⊂ V . This topology is known as the inductive limit of ∞ the topologies on C∞ K (V ), i.e. the topologies of CK (V ) are “pieced together” to form the topology of the union C∞ 0 (V ) (see [16]). The locally convex space C∞ 0 (V ) endowed with the inductive limit topology, is called the space of test functions and is commonly denoted by D(V ), in accordance with Schwartz’ notation in [17]. A distribution on V is a continuous linear functional on D(V ).
2. TEST FUNCTIONS AND DISTRIBUTIONS Throughout the sequel, Rn , n ∈ N represents the ndimensional Euclidean space. The following introduction to the theory of distributions is based on [4], [9] and [10], where further details about this subject can be found. 122
World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
Definition 2. The function G0 defined by
We shall denote the linear space of all distributions on V by D ′ (V ), the topological dual of D(V ). A function F defined on V is locally integrable on V if Z |F (x)|dx
G0 (x, y) :=
for all x, y ∈ R3 with x 6= y is a Green’s function for the operator −∇2 in the space R3 (see [6]).
E
is finite on every compact subset E of V and we shall use L1loc (V ) to denote the space of all locally integrable functions on V . All continuous functions on Rn , for example, are locally integrable, although some of them, such as polynomials, are not integrable on Rn . Clearly, 1
L (V ) ⊂
L1loc (V
1 4π|x − y|
Definition 3. For δ ∈ (0, 1), we may define the regularised Green’s function with respect to −∇2 for all x, y ∈ R3 by Gδ0 (x, y) :=
).
1 1
4π (|x − y|2 + δ) 2
.
Theorem 4. Let V be a bounded region in R3 and V represent the closure of V . Let f be a vector valued function, which is defined in the domain V such that f is continuous on V and has continuous first order derivatives in V . Then f = −∇ρ + ∇ × Λ with
If F ∈ L1loc (V ) then the linear functional TF defined on D(V ) by Z TF (φ) = F (x)φ(x)dx, φ ∈ D(V ) (1) V
(i)
is a distribution. A distribution T is said to be regular if there is a locally integrable function f on V such that Z F (x)φ(x)dx, φ ∈ D(V ). T (φ) =
∇x ρ = lim ∇x ρδ in D ′ (V ) δ→0+
with δ
ρ (x) =
V
Z
∇x Gδ0 (x, y) · f (y) dy
Z
∇x G0 (x, y) · f (y) dy
V
and
Otherwise it is singular. Among the topologies that can be defined on the vector space D ′ (V ), the most important is the one known as the weak topology. This is the locally convex topology defined by the family of seminorms
ρ(x) =
V
(ii) ∇x × Λ = lim ∇x × Λδ in D ′ (V ) δ→0+
Pφ (T ) = |T (φ)|
with
with φ ∈ D(V ) and T ∈ D ′ (V ), and it leads to the following definition of (weak) convergence in D ′ (V ). The sequence {Tk } in D ′ (V ) converges to 0 if and only if, for every φ ∈ D(V ), the sequence Tk (φ) converges to 0 in C. This is really a "pointwise" convergence on D(V ) and, as usual, we shall write
Λδ (x) =
Z
∇x Gδ0 (x, y) × f (y) dy
Z
∇x G0 (x, y) × f (y) dy
V
and Λ(x) =
V
for all x ∈ V , where Gδ0 is the regularised Green’s function for −∇2 on R3 and G0 is the Green’s function for −∇2 on R3 .
Tk → T in D ′ (V )
Proof. From Corollary 3.4 (i) of [2], we have
if the sequence (Tk − T ) converges to 0 in the sense of the above definition. The space of distributions D ′ (V ) is (sequentially) complete.
ρδ → ρ as δ → 0 + in D ′ (V ), i.e. the sequence of distributions Tρδ defined by Z Tρδ (φ) := ρδ (x)φ(x)dx
Theorem 1. If Tk ∈ D ′ (V ) for every k ∈ N and lim Tk = T, then lim ∂ α Tk = ∂ α T for every multi-index α ∈ Nn0 , where the derivatives are understood in the distributional sense.
V
converges to
3. INTERCHANGE OF LIMIT WITH DIFFERENTIAL AND INTEGRAL OPERATOR
Tρ (φ) :=
Z
V
123
ρ(x)φ(x)dx as δ → 0+
World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
for all φ in D(V ). Therefore, we can say Tρδ → Tρ as δ → 0+ in D ′ (V ). Due to Theorem 1 we have
and ∇x × Λδ
lim ∇x Tρδ = ∇x lim Tρδ = ∇x Tρ
δ→0+
in D ′ (V ). This implies that δ→0+
in D ′ (V ). This proves part (i). For (ii) once again, we use (ii) of Corollary 3.4 of [2], which gives
Theorem 5. ([11], p. 335) Let X be an open subset of Rm and Y be a measure space. Suppose that the function f : X × Y → R satisfies the following conditions: (i) f (x, y) is a measurable function of y for each x ∈ X. (x,y) exists for (ii) For almost all y ∈ Y , the derivative ∂f∂x i all x ∈ X. (iii) There is an integrable function G : Y → R such that ∂f (x,y) ∂xi ≤ G(y) for all x ∈ X. Then Z Z ∂ ∂ f (x, y)dy. f (x, y)dy = ∂xi Y ∂x i Y
Λ = lim Λδ in D ′ (V ). δ→0+
It means that the real components Λδ1 , Λδ2 and Λδ3 of Λδ converge to the corresponding real components Λ1 , Λ2 and Λ3 of Λ. To prove ∇x × Λ = lim ∇x × Λδ in D ′ (V ), δ→0+
we have to show that (∇x × Λ)i = lim (∇x × Λδ )i for i = 1, 2, 3 in D ′ (V ).
Theorem 6. Let f, ρδ , Λδ and Gδ0 be the same as in Theorem 4. Then
δ→0+
Here we show that (∇x × Λδ )1 → (∇x × Λ)1 in D ′ (V ). For the other components the proof is similar to this one. We know that
1. Z
∂ δ ∂ δ Λ − Λ (∇x × Λδ )1 = ∂x2 3 ∂x3 2
Z ∇x ∇x Gδ0 (x, y) · f (y) dy = ∇x ∇x Gδ0 (x, y)·f (y) dy
V
and
2. Z Z δ ∇x× ∇x G0 (x, y) × f (y) dy = ∇x× ∇x Gδ0 (x, y)×f (y) dy
Λδ2 → Λ2 in D ′ (V )
= ∇x × Λδ (x)
Λδ3 → Λ3 in D ′ (V ) therefore using Theorem 1 we have
Proof. 1. To prove the result, we have to show that (i) ∇x Gδ0 (x, y) · f (y) =: H(x, y) is a measurable function of y for each x ∈ V . (ii) For almost all y ∈ V , the expression ∇x H(x, y) exists for all x ∈ V . (iii) There is an integrable function G : V → R such that
∂ ∂ δ Λ → Λ2 in D ′ (V ) ∂x3 2 ∂x3 ∂ δ ∂ Λ → Λ3 in D ′ (V ). ∂x3 2 ∂x2
From the information given above, we get ∇x × Λδ 1 → (∇x × Λ)1 in D ′ (V ).
|∇x H(x, y)| ≤ G(y), for all x ∈ V . Due to Equations (11) and (12) of [2], we conclude that (i) is satisfied. Further, from Equation (A.1.5) of [1], p. 101, we obtain (ii). To show that (iii) is satisfied, we use
Similarly
2
V
V
and
V
= ∇x ρδ (x)
∂ ∂ (∇x × Λ)1 = Λ3 − Λ2 . ∂x2 ∂x3 As we know that
∇x × Λδ
→ (∇x × Λ)3 in D ′ (V ).
∇x × Λδ → (∇x × Λ) in D ′ (V ).
lim ∇x ρδ = ∇x lim ρδ = ∇x ρ
and
3
Combining all these we get
δ→0+
δ→0+
→ (∇x × Λ)2 in D ′ (V ) 124
World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
Equation (A.1.5) of [1], p. 101 as
get
∇x ∇x Gδ (x, y) · f (y) 2 = 1 × 0 (4π)2 !2 |x − y|2 + δ F1 (y) ((x − y) · f (y)) (x − y ) 1 1 −3 5 5 (|x − y|2 + δ) 2 (|x − y|2 + δ) 2 !2 |x − y|2 + δ F2 (y) ((x − y) · f (y)) (x2 − y2 ) + −3 5 5 (|x − y|2 + δ) 2 (|x − y|2 + δ) 2 !2 |x − y|2 + δ F3 (y) ((x − y) · f (y)) (x3 − y3 ) −3 + 5 5 2 2 (|x − y| + δ) (|x − y|2 + δ) 2 where f = (F1 , F2 , F3 )t . After simplying, we obtain
∇x × ∇x Gδ0 (x, y) × f (y) ≤ 2 ∇x × ∇x Gδ0 (x, y) × f (y) ≤ 3
δ + 28R2 5
2πδ 2 ×|f (y)| (5) δ + 28R2 5
2πδ 2 ×|f (y)|, (6)
where R is the radius of the ball containing V . From the discussion above, we can conclude that 2 √ ∇x × ∇x Gδ0 (x, y) × f (y) ≤ δ + 28R 3|f (y)| 5 2πδ 2 ˜ =: G(y). (7)
∇x
√ ∇x Gδ0 (x, y) · f (y) ≤ 3
δ + 16R
2
5
4πδ 2
=: G(y),
˜ is Since |f | is an integrable function of y, therefore, G also an integrable function of y. Hence, applying Theorem 5, we finally obtain 2.
|f (y)| (2)
where R is the radius of the ball containing V . Since f is an integrable function of y therefore G is also an integrable function of y. Hence, (iii) is satisfied. Finally, using Theorem 5 we get 1. 2. Due to Equation (17) of [2], we can say that ∇x Gδ0 (x, y) × f (y) is a measurable function of y for all x ∈ V . From Equations (A.1.8), (A.1.9) and (A.1.10) of [1], p. 103, we conclude that ∇x × ∇x Gδ0 (x, y) × f (y) exists for all x ∈ V . Further, from Equation (A.1.8) of [1], p. 103, we get
4. TENSORIAL KERNELS let c(V ) denote the set of continuous functions f : V → R3 and using the canonical orthonormal basis ε1 , ε2 , ε3 of R3 , a tensor f of rank-2 can be written as f=
i,j=1
5
+3
(|x − y|2 + δ) 2 |((x1 − y1 )F2 (y) − (x2 − y2 )F1 (y)) (x2 − y2 )| (|x − y|2 + δ) (|x − y|2 + δ)
5 2
or !
f > k :=
5
2πδ 2 ×|f (y)|,
f t (y)kt (., y) dy,
(10)
where f t (y) is the transpose of the vector f (y) and kt (x, y) is the transpose of the tensor k (x, y). As we use Lebesgue integrals therefore Z Z k (., y) f (y)dy. k (., y) f (y)dy =
(3)
δ + 28R2
Z
V
.
After simplifying, we have ∇x × ∇x Gδ0 (x, y) × f (y) ≤ 1
(8)
V
5 2
|((x3 − y3 )F1 (y) − (x1 − y1 )F3 (y)) (x3 − y3 )|
Fij εi ⊗ εj
where ⊗ is the dyadic product and Fij ∈ R. Clearly, f ∈ R3 ⊗ R3 . Let k : V × V → R3 ⊗ R3 and f ∈ c(V ), we define the convolution of f and k by Z k (., y) f (y)dy (9) k > f :=
∇x × ∇x Gδ0 (x, y) × f (y) ≤ 1 × 1 4π 2 2 |x − y| + δ |F1 (y)| +3
3 X
V
V
Theorem 7. Let kδ : V × V → R3 ⊗ R3 be a tensorial function defined by
(4)
where R is the radius of the ball containing V . Similarly, from Equations (A.1.9) and (A.1.10) of [1], p. 103, we
kδ (x, y) =
3 X
i,j=1
125
δ Kij (x, y) εi ⊗ εj
(11)
World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
for all x, y ∈ V and δ > 0. For i, j = 1, 2, 3, we define δ Kij : V × V → R as below 3 δ Kij (x, y) = 4π
1 3
(|x − y|2 + δ) 2
−
|x − y|2
5
(|x − y|2 + δ) 2
tively we get δ δ δ F1 (y)K11 (x, y) + F2 (y)K21 (x, y) + F3 (y)K31 (x, y)
!
=
,
=0
3F1 (y) 3
−3F2 (y)
if i = j and for all x, y ∈ V . If i 6= j we define δ Kij (x, y)
1 4π
−3F3 (y)
(12)
|x − y|2
− 3F1 (y)
5
(|x − y|2 + δ) 2 (|x − y|2 + δ) 2 (x1 − y1 )(x2 − y2 ) − (x1 − y1 )(x2 − y2 ) 5
(|x − y|2 + δ) 2
(x1 − y1 )(x3 − y3 ) − (x1 − y1 )(x3 − y3 ) 5
(|x − y|2 + δ) 2
!
.
After simplifying the equation given above, we get for all x, y ∈ V .
Let f be a vector valued function which is defined in the domain V such that f is continuous on V and has continuous first order derivatives in V . Then f > kδ → f as δ → 0+ in D ′ (V ).
×
Proof. From Equation (10) we have Z
f t (y)ktδ (x, y)dy (f > kδ ) (x) = V Z δ (F1 (y), F2 (y), F3 (y)) Kji (x, y) 3×3 dy, =
δ δ F1 (y)K11 (x, y) + F2 (y)K21 (x, y)
(13)
Let us investigate the components one by one. Using Equations (12) and (12) we obtain
3F1 (y) (|x − y|2 + δ)
3 2
− 3F1 (y)
|x − y|2
5
(|x − y|2 + δ) 2
!
.
δ δ δ F1 (y)K21 (x, y) + F2 (y)K22 (x, y) + F3 (y)K23 (x, y) δ = − ∇x ∇x G0 (x, y) · f (y) 2 (15) + ∇x × ∇x Gδ0 (x, y) × f (y) 2 . δ δ δ F1 (y)K13 (x, y) + F2 (y)K23 (x, y) + F3 (y)K33 (x, y) δ = − ∇x ∇x G0 (x, y) · f (y) 3 + ∇x × ∇x Gδ0 (x, y) × f (y) 3 . (16)
Putting the values of the components from Equations (14), (15) and (16) into Equation (13) we obtain Z (f > kδ ) (x) = − ∇x ∇x Gδ0 (x, y) · f (y) 1 V + ∇x × ∇x Gδ0 (x, y) × f (y) 1 , − ∇x ∇x Gδ0 (x, y) · f (y) 2 + ∇x × ∇x Gδ0 (x, y) × f (y) 2 , − ∇x ∇x Gδ0 (x, y) · f (y) 3 + ∇x × ∇x Gδ0 (x, y) × f (y) 3 dy.
δ δ δ F1 (y)K11 (x, y) + F2 (y)K21 (x, y) + F3 (y)K31 (x, y)
1 4π
((x3 − y3 )F1 (y) − (x1 − y1 )F3 (y)) (x3 − y3 )
δ δ δ F1 (y)K11 (x, y) + F2 (y)K21 (x, y) + F3 (y)K31 (x, y) δ = − ∇x ∇x G0 (x, y) · f (y) 1 (14) + ∇x × ∇x Gδ0 (x, y) × f (y) 1 .
δ +F3 (y)K31 (x, y), δ δ δ F1 (y)K12 (x, y) + F2 (y)K22 (x, y) + F3 (y)K32 (x, y), δ δ δ F1 (y)K13 (x, y) + F2 (y)K23 (x, y) + F3 (y)K33 (x, y) dy.
=
3 − 3 5 (|x − y|2 + δ) 2 (|x − y|2 + δ) 2 ((x3 − y3 )F3 (y)) (x1 − y1 ) 2F1 (y) −3 + 5 3 2 (|x − y| + δ) 2 (|x − y|2 + δ) 2 ((x1 − y1 )F2 (y) − (x2 − y2 )F1 (y)) (x2 − y2 ) +3 5 (|x − y|2 + δ) 2
Now, by comparing the equation given above with Equations (A.1.5) and (A.1.8) of [1], p. 101-103, we have
where Fi , i = 1, 2, 3 are the components of f and δ δ Kji (x, y) is a 3×3 matrix with components Kji (x, y). Now just performing the multiplication of matrices, we get
V
F1 (y)
−3
V
Z (f > kδ ) (x)=
1 4π ((x1 − y1 )F1 (y) + (x2 − y2 )F2 (y))
δ δ δ F1 (y)K11 (x, y) + F2 (y)K21 (x, y) + F3 (y)K31 (x, y) =
5
(|x − y|2 + δ) 2 +F2 (y) · 0 + F3 (y) · 0) .
2 −y2 ) Now adding and subtracting 3F2 (y) (x1 −y1 )(x and 5 2
(|x−y| +δ) 2
3 −y3 ) 3F3 (y) (x1 −y1 )(x in the third and the fourth term on 5 2
(|x−y| +δ) 2
the right hand side of the equation given above respec126
World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
The equation given above can be written as Z − ∇x ∇x Gδ0 (x, y) · f (y) (f > kδ ) (x) = V +∇x × ∇x Gδ0 (x, y) × f (y) dy Z ∇x ∇x Gδ0 (x, y) · f (y) dy = − Z V + ∇x × ∇x Gδ0 (x, y) × f (y) dy.
δ 10−2 10−3 10−4
Rooted mean square in reconstructed f 0.0300 0.0100 0.0056
Table 1: Rooted mean square error in the reconstruction of the synthetic vectorial function f for different values of the parameter δ.
V
Due to Theorem 6, the equation given above takes the form Z (f > kδ ) (x) = −∇x ∇x Gδ0 (x, y) · f (y)dy V Z ∇x Gδ0 (x, y) × f (y)dy +∇x ×
grid or a Gaussian grid. We use an equiangular grid and the quadrature theorem given in [7] and for the line integral we use the composite Simpson’s rule. Our vectorial synthetic function is given by f (y) = f (rη) = r3 ηY3,2 (η) sin(η2 ) cos(η3 )
V
y for y ∈ B1 (0). where r = |y| and η = (η1 , η2 , η3 ) = |y| Moreover, Y3,2 is a spherical harmonic of degree 3 and order 2. Analysing the graphs in Figures 3 to 4 and the errors in Table 1, we can say that we get a very good approximation of the vectorial function f . Acknowledgment. The author would like to thank the honorable referee for their valuable comments and suggestions which help us to finalize the presentation of this paper.
= −∇x ρδ (x) + ∇x × Λδ (x).
Taking the limit δ → 0+ on both hand sides, we get lim (f > kδ ) (x) = − lim ∇x ρδ (x)
δ→0+
δ→0+
+ lim ∇x × Λδ (x). δ→0+
Using Theorem 4 we can write the equation given above as follows lim (f > kδ ) (x)
δ→0+
=
−∇x ρ(x) + ∇x × Λ(x)
=
f (x).
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5. SOME NUMERICAL TESTS In this section, we present some numerical tests in which we reconstruct the synthetic vectorial function f by using the kernels given in Theorem 7 for different parameters δ = 0.01, δ = 0.001 and δ = 0.0001 We have also calculated the rooted mean square error for these values of δ which are given in Table 1. For all values of the parameter δ we used 100 × 256 × 256 grid points i.e. 100 point along radius, 256 points along longitude and 256 points along latitude for the integration on B1 (0) = V . To compute our convolution on V = B1 (0), we use the standard separation of an integral on B1 (0) Z
B1 (0)
F (x)dx
=
Z
0
1
r2
Z
(18)
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For the integral over the sphere, the most commonly used sampling measures have support on either an equiangular 127
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−1
0.1
0.15
0.2
0.25
Figure 1: (a) is the graph of the absolute values of the vector function f plotted in the y1 = 0 plane.
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0.06
−1
0.08
0.1
0.12
0.14
0.16
0.18
Figure 2: (a) is the graph of the absolute values of the reconstructed vector function f using the kernel of Theorem 7 with the parameter δ = 0.01. The function is plotted in the y1 = 0 plane.
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World Appl. Sci. J., 22 (Special Issue of Applied Math): 122-129, 2013
(a)
(a)
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 1
0 1 0.5
0.5
1
1
0.5
0
−0.5 −1
−1
0.02 0.04 0.06 0.08 0.1
0
−0.5
−0.5 −1
0.5
0
0
−0.5
0.12 0.14 0.16 0.18 0.2
0.22
0.05
−1
0.1
(b)
0.15
0.2
0.25
(b)
0.1
0.04
0.08
0.03
0.06 0.02 0.04 0.01
0.02 0 1
0 1 0.5
0.03
0.04
−0.5 −1
−1
0.05
0
−0.5
−0.5 −1
0.5
0
0
−0.5
0.02
1
0.5
0
0.01
0.5
1
0.06
0.07
0.08
0.09
0.005
Figure 3: (a) is the graph of the absolute values of the reconstructed vector function f using the kernel of Theorem 7 with the parameter δ = 0.001 and (b) is the graph of the absolute values of the difference of the actual vector function f and the reconstructed vector function f using the kernel of Theorem 7 with the parameter δ = 0.001. In both cases, the function is plotted in the y1 = 0 plane.
0.01
0.015
−1
0.02
0.025
0.03
0.035
Figure 4: (a) is the graph of the absolute values of the reconstructed vector function f using the kernel of Theorem 7 with the parameter δ = 0.0001 and (b) is the graph of the absolute values of the difference of the actual vector function f and the reconstructed vector function using the kernel of Theorem 7 with the parameter δ = 0.0001. In both cases, the function is plotted in the y1 = 0 plane.
129
