A new projector

Medeiros, J.C.O. de∗ Rodrigues dos Santos, S.†

Praciano-Pereira, T.‡

Departamento de MatematicaUniversidade Estadual Vale do Acarau

01/10/2003tarcisio@member.ams.org

pre-prints do Curso de Matematica de Sobralno. 2003.02

Editor Tarcisio Praciano-Pereiratarcisio@member.ams.org

Resumo

In this paper we present a new projector by modifying a previous

construction of an interpolation projector of ours. This new projector

produces quasi-convolution splines tangent to the target function f at

the precision points but they lacking continuity in the border of the sub-

intervals of the partition. This has been done by constructing a spline

approximation of f ′ and obtaining f by integration with multiple initial

conditions.

Some graphs are provided to compare the improvement of the ap-

proximation as the number of atoms in the base increases. The python

program which produces the graphics as well as make the statistics of er-

rors between some sample functions can be obtain with the authors. We

remark that the algorithm can parallelized.

Keywords: approximate identity, convolution regularization, convo-

lution splines, parallel programming, partition of the unity, projectors,

quasi-splines.

AMS-subject classification 41A63, 41A05, 41A15

∗Matematica - UVACe juan.medeiros@gmail.com†Matematica - UVACe‡Matematica - UFeRG-RS - tarcisio@member.ams.org

1 The organization of the paper

In the first section we make the construction of the new projector, in the secondone we prove the class of approximation we can reach with this new projectorand in the last part we present two graphics to show the improvement of thealgorithm along the increase of the number of atoms in the base.

2 Introduction: definitions and results.

Several authors have introduced convolution splines with which is possible toconstruct a partitition of the unity and a projector from a function space in aspline functions spaces.

We’ll start here with a partitition of the unity constructed in some way, let’ssay P = φi; i ∈ A is a partitition of the unity subordinated to an open coverC = {Ii; i ∈ A} of an interval [a, b]. We define, for an instance, using choice

axiom..., a collection of points of [a, b] to represent each member of C whichwill be called here the set of precision points P = {pi; i ∈ A}. We consider afinite open cover. And we consider the projector

P(f) =∑i∈A

f(pi)φi (1)

Obviously if f ∈ [P] the vector space generated by the elements of the par-titition of the unity, then P(f) = f which can proved by successive subtractiondue to our finiteness hypothesis. This proves that P is a projector. The factthat P is a partitition of the unity finishes the proof that P is an interpolationprojector.

See the figure (fig. 1) and the (fig. 2) to compare the improvement as thenumber of atoms increases. We observe that the algorithm takes roughly thesame time to process these two so disparate number of atoms.

3 The new projector

In the investigation which is the object of this paper we are considering thesummation

P(f) =∑i∈A

f ′(pi)φi ≈ f ′ (2)

i.e. putting f ′(pi) instead of f(pi) to have an approximation of f ′ and we retrivef, approximately, through the following algorithm:

• Given x ∈ [a, b] there exists finitely many non zero φi(x) whose sum is one,this finite number may be lesser than the cardinal of the partitition of theunity. It is furthermore possible to fix how many elements we have in this

2

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0

20

40

60

80

100

−10 −5 0 5 10

10 átomos − 2−splines − f(x)=(x+3)(x−3)sin(3x+5) [−10, 10]

’data’

Figura 1: 2-spline approximation f(x) = (x + 3)(x − 3)sin(3x + 5)x ∈ [−10, 10] using 10atoms

summation, in the computer algorithm we wrote, we are using exactlytwo.

• φi1, φik

;k∑

k=1

φik= 1 and consider the optimal real numbers α, β; [α, β] ⊃

∪k

k=1Iik

.

• Then

P(f)(x) = f(α) +

nk∑k=1

f ′(pik)

x∫

α

φik(t)dt

where x ∈ [α, β].

If the elements in the partitition of the unity are splines then P(f) is aspline. If f is a differentiable spline generated by P = φii ∈ A then P(f) = f

hence P is a projector. It remains to prove that P is an interpolation projectorwhich will be the point in the next section.

See here again two pictures to compare the result obtained with the newalgorithm, (fig. 3) and the (fig. 4)

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−10 −5 0 5 10

50 átomos − 2−splines − f(x)=(x+3)(x−3)sin(3x+5) [−10, 10]

’data’

Figura 2: 2-spline approximation f(x) = (x + 3)(x − 3)sin(3x + 5)x ∈ [−10, 10] using 50atoms

4 Estimate of the error

Referencias

[1] Rudin, W. Functional analysis - McGraw-Hill Book Company - 1973

[2] Delvos F-J, Schempp, W. Boolean methods in interpolation a approximation.

- Pitman research notes in Mathematics Series # 230 - 1989.

[3] Medeiros, J.C.O., Rodrigues dos Santos, S, Praciano-Pereira, T. Precision

of Convolution spline basis preprint - Matematica - UVA - 001/2002

http://www.uvanet.br/matematica/livros.php

[4] Medeiros, J.C.O., Rodrigues dos Santos, S, Praciano-Pereira, T. Existence

of compact support splines preprint - Matematica - UVA - 001/2003

http://www.uvanet.br/matematica/livros.php

[5] Lions, J.L. quelquer methodes de resolution des problemes aux limites non

lineaires Dunod/Gauthier Villars - Paris 1969

[6] Praciano-Pereira, T. Convolution spline basis preprint - presented at TenthInternational Conference on Approximation Theory - St. Louis, Missouri -March - 2001

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10 átomos − 2−splines − f(x)=(x+3)(x−3)sin(3x+5) [−10, 10]

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Figura 3: 2-quasi-spline approximation f(x) = (x + 3)(x − 3)sin(3x + 5)x ∈ [−10, 10] using10 atoms

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−10 −5 0 5 10

50 átomos − 2−splines − f(x)=(x+3)(x−3)sin(3x+5) [−10, 10]

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Figura 4: 2-quasi-spline approximation f(x) = (x + 3)(x − 3)sin(3x + 5)x ∈ [−10, 10] using50 atoms

5