Open access peer-reviewed chapter

Reproducing Kernel Functions

By Ali Akgül and Esra Karatas Akgül

Submitted: November 1st 2017Reviewed: February 12th 2018Published: May 23rd 2018

DOI: 10.5772/intechopen.75206

Downloaded: 297

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.

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×Eis called a reproducing kernel of the Hilbert space Hif and only if

  1. KtHfor all tE,

  2. φKt=φtfor all tEand all φH.

The last condition is called the reproducing property as the value of the function φat the point tis reproduced by the inner product of φwith 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 W2301are defined by

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

respectively. Thus, the space W2301is a reproducing kernel space, that is, for each fixed y01and any vW2301,there exists a function Rysuch 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 T2301are defined by

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

respectively. The space T2301is a reproducing kernel Hilbert space, and its reproducing kernel function rsis 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 G2101are defined by

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

respectively. The space G2101is a reproducing kernel space, and its reproducing kernel function Qyis given by [1] as

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

Theorem 1.1. The space W2301is a complete reproducing kernel space whose reproducing kernel Ryis 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 ciyand diyi=126can be obtained. Thus, Ryis 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 WΩare defined by

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

respectively.

Theorem 1.2. The WΩis 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 ŴΩare defined by [1] as

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

respectively. ŴΩis a reproducing kernel space, and its reproducing kernel function Gysis

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 W2301are defined, respectively, by

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

and

uW23=uuW23,uW2301.

The space W2301is a reproducing kernel space, that is, for each fixed y01and any uxW2301,there exists a function Ryxsuch that

uy=uxRyxW23.

Definition 1.4.

W2101=uxux,is absolutely continuous in01uxL201,x01,

The inner product and the norm in W2101are defined, respectively, by

uxɡxW21=u0ɡ0+01uxɡxdx,ux,ɡxW2101,E28

and

uW21=uuW21,uW2101.E29

The space W2101is a reproducing kernel space, and its reproducing kernel function Txyis given by

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

Theorem 1.5. The space W2301is a complete reproducing kernel space, and its reproducing kernel function Ryxcan 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 ciyand diyi=126can be obtained. Thus Ryxis 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 W2401are defined, respectively, by

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

The space W2401is a reproducing kernel space, that is, for each fixed.

y01and any vxW2401,there exists a function Ryxsuch that

vy=vxRyxW24E39

Similarly, we define the space

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

The inner product and the norm in W220Tare defined, respectively, by

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

Thus, the space W220Tis also a reproducing kernel space, and its reproducing kernel function rstcan 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 W2201are defined, respectively, by

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

The space W2201is a reproducing kernel space, and its reproducing kernel function Qyxis given by

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

Similarly, the space W210Tis defined by

W210T=vtvtis absolutely continuous in0T,vtL20T,t0TE46

The inner product and the norm in W210Tare defined, respectively, by

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

The space W210Tis a reproducing kernel space, and its reproducing kernel function qstis given by

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

Further, we define the space WΩas

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

and the inner product and the norm in WΩare 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 W2401is a complete reproducing kernel space, and its reproducing kernel function Ryxcan 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 ciyve diyi=128can be obtained. Thus, Ryxis given by

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

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Ali Akgül and Esra Karatas Akgül (May 23rd 2018). Reproducing Kernel Functions, Differential Equations - Theory and Current Research, Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.75206. Available from:

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