Open access peer-reviewed chapter

Reproducing Kernel Functions

Written By

Ali Akgül and Esra Karatas Akgül

Submitted: 01 November 2017 Reviewed: 12 February 2018 Published: 23 May 2018

DOI: 10.5772/intechopen.75206

From the Edited Volume

Differential Equations - Theory and Current Research

Edited by Terry E. Moschandreou

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Abstract

In this chapter, we obtain some reproducing kernel spaces. We obtain reproducing kernel functions in these spaces. These reproducing kernel functions are very important for solving ordinary and partial differential equations.

Keywords

  • reproducing kernel functions
  • reproducing kernel spaces
  • ordinary and partial differential equations

1. Introduction

Reproducing kernel spaces are special Hilbert spaces. These spaces satisfy the reproducing property. There is an important relation between the order of the problems and the reproducing kernel spaces.

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2. Reproducing kernel spaces

In this section, we define some useful reproducing kernel functions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Definition 2.1 (reproducing kernel). Let E be a nonempty set. A function K:E×E is called a reproducing kernel of the Hilbert space H if and only if

  1. KtH for all tE,

  2. φKt=φt for all tE and all φH.

The last condition is called the reproducing property as the value of the function φ at the point t is reproduced by the inner product of φ with Kt.

Then, we need some notation that we use in the development of this chapter. Next, we define several spaces with inner product over those spaces. Thus, the space defined as

W2301=vvvv:01Rareabsolutely continuousv3L201E1

is a Hilbert space. The inner product and the norm in W2301 are defined by

vɡW23=i=02vi0ɡi0+01v3xɡ3xdx,v,ɡW2301,vW23=vvW23,vW2301,E2

respectively. Thus, the space W2301 is a reproducing kernel space, that is, for each fixed y01 and any vW2301, there exists a function Ry such that

vy=vxRyxW23,E3

and similarly, we define the space

T2301=vv,v,v:01Rareabsolutely continuous,vL201,v0=0,v0=0E4

The inner product and the norm in T2301 are defined by

vɡT23=i=02vi0ɡi0+01v′′′tɡ′′′tdt,v,ɡT2301,vT23=vvT23,vT2301,E5

respectively. The space T2301 is a reproducing kernel Hilbert space, and its reproducing kernel function rs is given by [1] as

rs=14s2t2+112s2t3124st4+1120t5,ts,14s2t2+112s3t2124ts4+1120s5,t>s,E6

and the space

G2101=vv:01Ris absolutely continuousvxL2[01],E7

is a Hilbert space, where the inner product and the norm in G2101 are defined by

vɡG21=vi0ɡi0+01vxɡxdx,v,ɡG2101,vG21=vvG21,vG2101,E8

respectively. The space G2101 is a reproducing kernel space, and its reproducing kernel function Qy is given by [1] as

Qy=1+x,xy1+y,x>y.E9

Theorem 1.1. The space W2301 is a complete reproducing kernel space whose reproducing kernel Ry is given by

Ryx=i=16ciyxi1,xy,i=16diyxi1,x>y,E10

where

c1y=1,c2y=y,c3y=y24,c4y=y212,c5y=124y,c6y=1120,d1y=1+y5120,d2y=y424+y,d3y=y24+y312,d4y=d5y=d6y=0.

Proof. Since

vRyW23=i=02vi0Ryi0+01v3xRy3xdx,(v,RyW2301E11

through iterative integrations by parts for (11), we have

vxRyxW24=i=02vi0Ryi012iRy5i0+i=0212ivi1Ry5i1+01vxRy6xdx.E12

Note, the property of the reproducing kernel as

vxRyxW23=vy.E13

If

Ry0Ry50=0,Ry0+Ry40=0,Ry0Ry0=0,Ry31=0,Ry41=0,Ry51=0,E14

Then by (11), we obtain

Ry6x=δxy,E15

when xy,

Ry6x=0,E16

therefore,

Ryx=i=16ciyxi1,xy,i=16diyxi1,x>y,E17

Since

Ry6x=δxy,E18

we have

kRy+y=kRyy,k=0,1,2,3,4,5Ry+y5Ryy=1.E19

From (14) and (19), the unknown coefficients ciy and diy i=126 can be obtained. Thus, Ry is given by

Ry=1+yx+14y2x2+112y2x3124yx4+1120x5,xy1+yx+14y2x2+112y3x2124xy4+1120y5,x>y.E20

Now, we note that the space given in [1] as

WΩ=vxt4vx2t2,is completely continuous inΩ=01×01,6vx3t3L2Ω,vx0=0,vx0t=0E21

is a binary reproducing kernel Hilbert space. The inner product and the norm in WΩ are defined by

vxtɡxtW=i=02013t3ixiv0t3t3ixiɡ(0t)dt+j=02jtjvx0jtjɡ(x0)W23+01013x33t3vxt3x33t3ɡ(xt)dxdt,vw=vvW,vWΩ,E22

respectively.

Theorem 1.2. The WΩ is a reproducing kernel space, and its reproducing kernel function is

Kys=RyrsE23

such that for any vWΩ,

vys=vxtKysxtWKysxt=Kxtys.E24

Similarly, the space

ŴΩ=vxtv(xt)is completely continuous inΩ=[01]×[01]2vxtL2ΩE25

is a binary reproducing kernel Hilbert space. The inner product and the norm in ŴΩ are defined by [1] as

vxtɡxtŴ=01tv0ttɡ(0t)dt+vx0ɡx0W21+0101xtvxtxtɡ(xt)dxdt,vŴ=vvŴ,vŴΩ,E26

respectively. ŴΩ is a reproducing kernel space, and its reproducing kernel function Gys is

Gys=QyQs.E27

Definition 1.3.

W2301=uxux,ux,ux,areabsolutely continuous in01u3xL201,x01,u0=0,u1=0.

The inner product and the norm in W2301 are defined, respectively, by

uxɡxW23=i=02ui0ɡi0+01u3xɡ3xdx,ux,ɡxW2301

and

uW23=uuW23,uW2301.

The space W2301 is a reproducing kernel space, that is, for each fixed y01 and any uxW2301, there exists a function Ryx such that

uy=uxRyxW23.

Definition 1.4.

W2101=uxux,is absolutely continuous in01uxL201,x01,

The inner product and the norm in W2101 are defined, respectively, by

uxɡxW21=u0ɡ0+01uxɡxdx,ux,ɡxW2101,E28

and

uW21=uuW21,uW2101.E29

The space W2101 is a reproducing kernel space, and its reproducing kernel function Txy is given by

Txy=1+x,xy,1+y,x>y.E30

Theorem 1.5. The space W2301 is a complete reproducing kernel space, and its reproducing kernel function Ryx can be denoted by

Ryx=i=16ciyxi1,xy,i=16diyxi1,x>y,

where

c1y=0,c2y=5516y41156y5526y2578y3+313y,c3y=5624y41624y5+21104y25312y3526y,c4y=51872y411872y5+7104y25936y3578y,c5y=53744y4+13744y5+5624y2+51872y31104y,c6y=1120+13744y4118720y51624y211872y31156y,d1y=1120y5,d2y=1104y41156y5526y2578y3+313y,d3y=5624y41624y5+21104y2+7104y3526y,d4y=51872y411872y55312y25936y3578y,d5y=53744y4+13744y5+5624y2+51872y3+5156y,d6y=1156y+13744y4118720y51624y211872y3.

Proof. We have

uxRyxW23=i=02ui0Ryi0+01u3xRy3xdx.E31

Through several integrations by parts for (31), we have

uxRyxW26=i=02ui0Ryi012iRy5i0+i=0212iui1Ry5i101uxRy6xdx.E32

Note that property of the reproducing kernel

uxRyxW23=uy,

If

Ry'0Ry30=0,Ry'0+Ry40=0,Ry31=0,Ry41=0,E33

then by (31), we have the following equation:

Ry6x=δxy,
when xy,
Ry6x=0,

therefore,

Ryx=i=16ciyxi1,xy,i=16diyxi1,x>y,

Since

Ry6x=δxy,

we have

kRy+y=kRyy,k=0,1,2,3,4,E34

and

5Ry+y5Ryy=1.E35

Since RyxW2301, it follows that

Ry0=0,Ry1=0,E36

From (33)–(36), the unknown coefficients ciy and diy i=126 can be obtained. Thus Ryx is given by

Ryx=5516xy41156xy5526xy2578xy3+313xy+5624x2y41624x2y5+21104x2y25312x2y3526x2y+51872x3y411872x3y5+7104x3y25936x3y3578x3y53744x4y4+13744x4y5+5624x4y2+51872x4y31104x4y1156x5y+13744x5y4118720x5y51624x5y211872x5y3,xy5516yx41156yx5526yx2578yx3+313xy+5624y2x41624y2x5+21104x2y25312y2x3526y2x+51872y3x411872y3x5+7104y3x25936x3y3578y3x53744x4y4+13744y4x5+5624y4x2+51872y4x31104y4x1156y5x+13744y5x4118720x5y51624y5x211872y5x3,x>y
W2401=vxvx,vx,v′′x,v′′′xareabsolutely continuous in01,v4xL201,x01E37

The inner product and the norm in W2401 are defined, respectively, by

vxɡxW24=i=03vi0ɡi0+01v4xɡ4xdx,vx,ɡxW2401,vW24=vvW24,vW2401.E38

The space W2401 is a reproducing kernel space, that is, for each fixed.

y01 and any vxW2401, there exists a function Ryx such that

vy=vxRyxW24E39

Similarly, we define the space

W220T=vtvt,vtareabsolutely continuous in0T,vtL20T,t0T,v0=0E40

The inner product and the norm in W220T are defined, respectively, by

vtɡtW22=i=01vi0ɡi0+0Tv′′tɡ′′tdt,vt,ɡtW220T,vW1=vvW22,vW220T.E41

Thus, the space W220T is also a reproducing kernel space, and its reproducing kernel function rst can be given by

rst=st+s2t216t3,ts,st+t2s216s3,t>s,E42

and the space

W2201=vxvx,vxareabsolutely continuous in01,v′′xL201,x01E43

where the inner product and the norm in W2201 are defined, respectively, by

vtɡtW22=i=01vi0ɡi0+0Tv′′tɡ′′tdt,vt,ɡtW2201,vW2=vvW22,vW2201.E44

The space W2201 is a reproducing kernel space, and its reproducing kernel function Qyx is given by

Qyx=1+xy+y2x216x3,xy,1+xy+x2y216y3,x>y.E45

Similarly, the space W210T is defined by

W210T=vtvtis absolutely continuous in0T,vtL20T,t0TE46

The inner product and the norm in W210T are defined, respectively, by

vtɡtW21=v0ɡ0+0Tvtɡtdt,vt,ɡtW210T,vW21=vvW21,vW210T.E47

The space W210T is a reproducing kernel space, and its reproducing kernel function qst is given by

qst=1+t,ts,1+s,t>s.E48

Further, we define the space WΩ as

WΩ=vxt4vx3t,is completely continuous,inΩ=01×0T,6vx4t2L2Ω,vx0=0E49

and the inner product and the norm in WΩ are defined, respectively, by

vxtɡxtW=i=030T2t2ixiv0t2t2ixiɡ(0t)dt+j=01jtjvx0jtjɡ(x0)W24+0T014x42t2vxt4x42t2ɡ(xt)dxdt,vW=vvW,vWΩ.E50

Now, we have the following theorem:

Theorem 1.6. The space W2401 is a complete reproducing kernel space, and its reproducing kernel function Ryx can be denoted by

Ryx=i=18ciyxi1,xy,i=18diyxi1,x>y,E51

where

c1y=1,c2y=y,c3y=14y2,c4y=136y3,c5y=1144y3,c6y=1240y2,c7y=1720y,c8y=15040,d1y=115040y7,d2y=y+1720y6,d3y=14y21240y5,d4y=136y3+1144y4,d5y=0,d6y=0,d7y=0,d8y=0.E52

Proof. Since

vxRyxW24=i=03vi0Ryi0+01v4xRy4xdx,vxRyxW2401E53

through iterative integrations by parts for (53), we have

vxRyxW24=i=03vi0Ryi013iRy7i0+i=0313ivi1Ry7i1+01vxRy8xdx.E54

Note that property of the reproducing kernel

vxRyxW24=vy.E55

If

Ry0+Ry70=0,Ry0Ry60=0,Ry0+Ry50=0,Ry0Ry40=0,Ry41=0,Ry51=0,Ry61=0,Ry71=0,E56

then by (54), we obtain the following equation:

Ry8x=δxy,E57

when xy,

Ry8x=0;E58

therefore,

Ryx=i=18ciyxi1,xy,i=18diyxi1,x>y.E59

Since

Ry8x=δxy,E60

we have

kRy+y=kRyy,k=0,1,2,3,4,5,6,E61
7Ry+y7Ryy=1.E62

From (56)–(62), the unknown coefficients ciy ve diyi=128 can be obtained. Thus, Ryx is given by

Ryx=1+yx+14y2x2+136y3x3+1144y3x41240y2x5+1720yx615040x7,xy,1+xy+14x2y2+136x3y3+1144x3y41240x2y5+1720xy615040y7,x>y.E63

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Written By

Ali Akgül and Esra Karatas Akgül

Submitted: 01 November 2017 Reviewed: 12 February 2018 Published: 23 May 2018