Bernstein-Durrmeyer operators with arbitrary weight functions

transcript

Bernstein-Durrmeyer operatorswith arbitrary weight functions

Elena E. Berdysheva

German University of Technology in Oman

Muscat, Sultanate of Oman

Bernstein-Durrmeyer operators with arbitrary weight functions – p. 1/32

Bernstein basis polynomialsStandard simplex in Rd:

Sd := {x = (x1, . . . , xd) ∈ R

0 ≤ x1, . . . , xd ≤ 1, x1 + · · ·+ xd ≤ 1}.

Sd := {x = (x1, . . . , xd) ∈ R

0 ≤ x1, . . . , xd ≤ 1, x1 + · · ·+ xd ≤ 1}.

Barycentric coordinates:

x = (x0, x1, . . . , xd), x0 := 1− x1 − · · · − xd.

Sd := {x = (x1, . . . , xd) ∈ R

0 ≤ x1, . . . , xd ≤ 1, x1 + · · ·+ xd ≤ 1}.

Barycentric coordinates:

x = (x0, x1, . . . , xd), x0 := 1− x1 − · · · − xd.

The d-variate Bernstein basis polynomials of degree n aredefined by

Bα(x) :=

α0!α1! . . . αd!xα0

0 xα1

1 · · · xαd

α = (α0, α1, . . . , αd) ∈ Nd+10 with |α| := α0 + α1 + · · ·+ αd = n.

Bernstein basis polynomialsOne-dimensional case: n ∈ N and k = 0, 1, . . . , n,

pn,k(x) :=

xk (1− x)n−k, x ∈ [0, 1].

pn,k(x) :=

xk (1− x)n−k, x ∈ [0, 1].

0.2 0.4 0.6 0.8 1x

p3,0 = (1− x)3

pn,k(x) :=

xk (1− x)n−k, x ∈ [0, 1].

0.2 0.4 0.6 0.8 1x

p3,0 = (1− x)3

p3,1 = 3 x (1− x)2

pn,k(x) :=

xk (1− x)n−k, x ∈ [0, 1].

0.2 0.4 0.6 0.8 1x

p3,0 = (1− x)3

p3,1 = 3 x (1− x)2

p3,2 = 3 x2 (1− x)

pn,k(x) :=

xk (1− x)n−k, x ∈ [0, 1].

0.2 0.4 0.6 0.8 1x

p3,0 = (1− x)3

p3,1 = 3 x (1− x)2

p3,2 = 3 x2 (1− x)

p3,3 = x3

Bernstein basis polynomials

Clearly,∑

|α|=n

Bα(x) = 1.

Clearly,∑

|α|=n

Bα(x) = 1.

The polynomials {Bα}|α|=n constitute a basis of the spaceof d-variate algebraic polynomials of total degree ≤ n.

Clearly,∑

|α|=n

Bα(x) = 1.

The Bernstein operator is defined for f ∈ C(Sd) by(

(x) :=∑

|α|=n

Bα(x).

Clearly,∑

|α|=n

Bα(x) = 1.

The Bernstein operator is defined for f ∈ C(Sd) by(

(x) :=∑

|α|=n

Bα(x).

This is a positive linear operator that reproduces linearfunctions.

Uniform convergence for every function in C(Sd).

Bernstein-Durrmeyer operatorA similar construction for integrable functions?

Definition. The Bernstein-Durrmeyer operator is definedfor f ∈ Lq(Sd), 1 ≤ q < ∞, or f ∈ C(Sd) by

(x) :=∑

|α|=n

Sdf(y) Bα(y) dy∫

SdBα(y) dy

Bα(x).

(x) :=∑

|α|=n

Sdf(y) Bα(y) dy∫

SdBα(y) dy

Bα(x).

Mn is a positive linear operator that reproduces constantfunctions. Convergence in Lq(Sd), 1 ≤ q < ∞, and in C(Sd).

(x) :=∑

|α|=n

Sdf(y) Bα(y) dy∫

SdBα(y) dy

Bα(x).

Introduced in the one-dimensional case by Durrmeyer(1967) and, independently, by Lupas (1972).

(x) :=∑

|α|=n

Sdf(y) Bα(y) dy∫

SdBα(y) dy

Bα(x).

Introduced in the one-dimensional case by Durrmeyer(1967) and, independently, by Lupas (1972). Becameknown due to Derriennic (starting from 1981).

(x) :=∑

|α|=n

Sdf(y) Bα(y) dy∫

SdBα(y) dy

Bα(x).

Introduced in the one-dimensional case by Durrmeyer(1967) and, independently, by Lupas (1972). Becameknown due to Derriennic (starting from 1981).Extension to functions on the d-dimensional simplex:Derriennic (starting from 1985).

Weighted Bernstein-Durrmeyer operator

Let ρ be a non-negative bounded (regular) Borel measureon Sd such that supp (ρ) \ (∂Sd) 6= ∅.

Lqρ(S

d), 1 ≤ q < ∞: the weighted Lq-space with the norm

‖f‖Lqρ:=

|f(x)|q dρ(x)

Lqρ(S

‖f‖Lqρ:=

|f(x)|q dρ(x)

Definition. The Bernstein-Durrmeyer operator with respectto the measure ρ is defined for f ∈ L

d), 1 ≤ q < ∞, orf ∈ C(Sd) by

(Mn,ρ f)(x) :=∑

|α|=n

Sdf(y) Bα(y) dρ(y)∫

SdBα(y) dρ(y)

Bα(x).

Lqρ(S

‖f‖Lqρ:=

|f(x)|q dρ(x)

Definition. The Bernstein-Durrmeyer operator with respectto the measure ρ is defined for f ∈ L

d), 1 ≤ q < ∞, orf ∈ C(Sd) by

(Mn,ρ f)(x) :=∑

|α|=n

Sdf(y) Bα(y) dρ(y)∫

SdBα(y) dρ(y)

Bα(x).

Mn,ρ is a positive linear operator that reproduces constantfunctions.

Jacobi weightsThe weighted Bernstein-Durrmeyer operator Mn,ρ is verywell studied for Jacobi weights, i.e., for

dρ(x) = xµ dx,

with µ = (µ0, µ1, . . . , µd) ∈ Rd+1, where µi > −1,i = 0, 1, . . . , d.

dρ(x) = xµ dx,

Bernstein-Durrmeyer operators with respect to Jacobiweights were introduced by Paltanea (1983), Berens andXu (1991), in the multivariate case by Ditzian (1995).

dρ(x) = xµ dx,

Bernstein-Durrmeyer operators with respect to Jacobiweights were introduced by Paltanea (1983), Berens andXu (1991), in the multivariate case by Ditzian (1995).

They were studied by many authors, e.g., Derriennic,Berens, Xu, Ditzian, Chen, Ivanov, X.-L. Zhou, Knoop,Gonska, Heilmann, Abel, Jetter, Stöckler, . . .

Jacobi weightsBerens and Xu noticed that the Bernstein-Durrmeyeroperator with respect to Jacobi weight is a summationmethod for Jacobi series.

This follows from the spectral properties of theBernstein-Durrmeyer operators with respect to Jacobiweights:

This follows from the spectral properties of theBernstein-Durrmeyer operators with respect to Jacobiweights: the eigenfunctions in this case are Jacobipolynomials.

For d = 1, µ1 = µ0 = −12 , it is exactly the de la

Vallée-Poussin mean for Chebyshev series (1908).

The univariate ultraspherical case (µ1 = µ0) was studiedalready by Kogbetliantz (1922),

The univariate ultraspherical case (µ1 = µ0) was studiedalready by Kogbetliantz (1922),the general univariate case by Bavinck (1976).

Motivation: Learning TheoryIn a joint paper with Kurt Jetter (JAT, 2010), we started tostudy the multivariate Bernstein-Durrmeyer operators withrespect to general measure.

We were motivated by paperD.-X. Zhou and K. Jetter, Approximation withpolynomial kernels and SVM classifiers, Adv.Comput. Math. 25 (2006), 323-344.

They considered the univariate Mn,ρ and used it forestimates for SVM classifiers.

Recently (2012), Bing-Zheng Li used the multivariateoperators Mn,ρ to obtain estimates for learning rates ofleast-square regularized regression with polynomialkernels.

Motivation: Learning TheoryTypical problems in learning theory:

Regression. E.g., least squares.

–2 –1 1 2 3x

Motivation: Learning TheoryTypical problems in learning theory:

Regression. E.g., least squares.

–2 –1 1 2 3x

Classification.

Convergence: examplesNumerical experiments show that convergence holds for awide class of measures ρ.

Example 1. Consider the measure dρ(x) = w(x) dx with

w(x) =

x− 1

0.2 0.4 0.6 0.8 1

Example 1

dρ(x) =∣

x− 12

)2 − 18

∣dx, f(x) = x, n = 5 :

0.2 0.4 0.6 0.8 1

xBernstein-Durrmeyer operators with arbitrary weight functions – p. 15/32

Example 1

dρ(x) =∣

x− 12

)2 − 18

∣dx, f(x) = x, n = 20 :

0.2 0.4 0.6 0.8 1

Example 1

dρ(x) =∣

x− 12

)2 − 18

∣dx, f(x) = x, n = 100 :

0.2 0.4 0.6 0.8 1

Example 1

dρ(x) =∣

x− 12

)2 − 18

∣dx, f(x) = x, n = 500 :

0.2 0.4 0.6 0.8 1

Convergence: examplesOn the other hand, it is not difficult to construct an exampleof an operator for which convergence in C([0, 1]) fails.

Example 2. dρ(x) = w(x) dx with

w(x) =

1, 0 ≤ x ≤ 12 ,

0, 12 < x ≤ 1.

0.2 0.4 0.6 0.8 1

w(x) =

1, 0 ≤ x ≤ 12 ,

0, 12 < x ≤ 1.

0.2 0.4 0.6 0.8 1

The Bernstein-Durrmeyer operator has the form

(Mn,ρ f)(x) =

0 f(y) yk (1− y)n−k dy∫

0 yk (1− y)n−k dy

xk (1− x)n−k.

w(x) =

1, 0 ≤ x ≤ 12 ,

0, 12 < x ≤ 1.

0.2 0.4 0.6 0.8 1

The Bernstein-Durrmeyer operator has the form

(Mn,ρ f)(x) =

0 f(y) yk (1− y)n−k dy∫

0 yk (1− y)n−k dy

xk (1− x)n−k.

Then for f(x) = x we have (Mn,ρ f)(x) ≤ 12 , x ∈ [0, 1].

Example 2dρ = χ[0, 12 ]

dx, f(x) = x, n = 5 :

0.2 0.4 0.6 0.8 1

dx, f(x) = x, n = 15 :

0.2 0.4 0.6 0.8 1

dx, f(x) = x, n = 30 :

0.2 0.4 0.6 0.8 1

dx, f(x) = x, n = 100 :

0.2 0.4 0.6 0.8 1

dx, f(x) = x, n = 200 :

0.2 0.4 0.6 0.8 1

dx, f(x) = x, n = 200 :

0.2 0.4 0.6 0.8 1

Convergence in C(Sd)We give necessary and sufficient conditions on themeasure ρ for convergence in C(Sd).

Recall that a measure ρ on Sd is called strictly positive, ifρ(A ∩ Sd) > 0 for every open set A ⊂ Rd with A ∩ Sd 6= ∅.

This is equivalent to the fact that supp (ρ) = Sd.

Theorem. (EB, JMAA, 2012) We have

limn→∞

‖f −Mn,ρ f‖C = 0 for all f ∈ C(Sd)

if and only if ρ is strictly positive on Sd.

Convergence in C(Sd)We say that ρ is a Jacobi-like measure, if

dρ(x) = w(x) dx,

and there are two exponents ν ≥ µ > −e and two constants0 < a,A < ∞ such that

a xν ≤ w(x) ≤ A x

µ, x ∈ Sd.

dρ(x) = w(x) dx,

and there are two exponents ν ≥ µ > −e and two constants0 < a,A < ∞ such that

a xν ≤ w(x) ≤ A x

µ, x ∈ Sd.

Denote ϕei(x) = xi, ϕ2ei

(x) = x2i .

Convergence in C(Sd)Theorem. (Kurt Jetter-EB, JAT, 2010) Let ρ be aJacobi-like measure with |ν| − |µ| < 1. Then

‖ϕei−Mn,ρ(ϕei

)‖C ≤ C n−1−(|ν|−|µ|)

and‖ϕ2ei

−Mn,ρ(ϕ2ei)‖C ≤ C n−

1−(|ν|−|µ|)2 .

Convergence in C(Sd)Theorem. (Kurt Jetter-EB, JAT, 2010) Let ρ be aJacobi-like measure with |ν| − |µ| < 1. Then

‖ϕei−Mn,ρ(ϕei

)‖C ≤ C n−1−(|ν|−|µ|)

and‖ϕ2ei

−Mn,ρ(ϕ2ei)‖C ≤ C n−

1−(|ν|−|µ|)2 .

Corollary. Let ρ be a Jacobi-like measure with |ν| − |µ| < 1.Let f ∈ C(Sd). Then

‖f −Mn,ρ f‖C ≤ C ω(

f, n−1−(|ν|−|µ|)

where ω(f, δ) = sup {|f(x)− f(t)| : ‖t− x‖2 < δ} denote themodulus of continuity of f .

Convergence on supp ρ

Theorem. (EB, 2012) Let x ∈ (supp ρ)◦. Let f be boundedon supp ρ and continuous at x. Then

limn→∞

|f(x)−Mn,ρ f(x)| = 0.

Convergence on supp ρ

Theorem. (EB, 2012) Let x ∈ (supp ρ)◦. Let f be boundedon supp ρ and continuous at x. Then

limn→∞

|f(x)−Mn,ρ f(x)| = 0.

Theorem. (EB, 2012) Let A be a compact set,A ⊂ (supp ρ)◦. Let f be bounded on supp ρ and continuouson A. Then

limn→∞

‖f −Mn,ρ f‖C(A) = 0.

Convergence in Lqρ(S

Theorem. (Bing-Zheng Li, 2012) Let ρ be a non-negativebounded (regular) Borel measure on Sd such thatsupp (ρ) \ (∂Sd) 6= ∅. Let 1 ≤ q < ∞. Then

limn→∞

‖f −Mn,ρ f‖Lqρ= 0

for every f ∈ Lqρ(S

Consider the K-functional

Kq(f, t) = inf {‖f − g‖Lqρ+ t max

i=1,...,d‖∂ig‖C : g ∈ C1(Sd)}.

Theorem. (Bing-Zheng Li-EB, 2012) Let ρ be anon-negative bounded Borel measure on Sd such thatsupp ρ \ ∂Sd 6= ∅, and let f ∈ L

d), 1 ≤ q < ∞. Then

‖f −Mn,ρ f‖Lqρ≤ 2Kq

f,Cq√nd[

ρ(Sd)]

, 1 ≤ q < ∞,

where Cq is a constant that depends only on q. Moreover,Cq = 1 for 1 ≤ q ≤ 2.

i=1,...,d‖∂ig‖C : g ∈ C1(Sd)}.

Theorem. (Bing-Zheng Li-EB, 2012) Let ρ be anon-negative bounded Borel measure on Sd such thatsupp ρ \ ∂Sd 6= ∅, and let f ∈ L

d), 1 ≤ q < ∞. Then

‖f −Mn,ρ f‖Lqρ≤ 2Kq

f,Cq√nd[

ρ(Sd)]

, 1 ≤ q < ∞,

where Cq is a constant that depends only on q. Moreover,Cq = 1 for 1 ≤ q ≤ 2.

Remark: the cases q = 1, q = 2 are due to Bing-Zheng Li.

Thank you for your attention!

Bernstein-Durrmeyer operators with arbitrary weight functions

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