Reproducing Kernel Functions

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.

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 $K:E\times E\to \mathrm{ℂ}$ is called a reproducing kernel of the Hilbert space $H$ if and only if

1. $K\left(\cdot ,t\right)\in H$ for all $t\in E,$

2. $⟨\phi ,K\left(\cdot ,t\right)⟩=\phi \left(t\right)$ for all $t\in E$ and all $\phi \in H.$

The last condition is called the reproducing property as the value of the function $\phi$ at the point $t$ is reproduced by the inner product of $\phi$ with $K\left(\cdot ,t\right).$

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

$$W_2^3\left[0,1\right]=\left\{v|v,v^{\prime },{v^{\prime }}^{\prime }:\left[0,1\right]\to \mathbb{R}\mathrm{are}\text{absolutely\hspace{0.17em}continuous},v^{\left(3\right)}\in L^2\left[0,1\right]\right\}$$E1

is a Hilbert space. The inner product and the norm in $W_2^3\left[0,1\right]$ are defined by

$$\begin{array}{c}{⟨v,ɡ⟩}_{W_2^3}=\sum _{i=0}^2{v}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\left(3\right)}\left(x\right){ɡ}^{\left(3\right)}\left(x\right)\mathrm{d}x,v,ɡ\in W_2^3\left[0,1\right],\\ \parallel v{\parallel }_{W_2^3}=\sqrt{{⟨v,v⟩}_{W_2^3}},v\in W_2^3\left[0,1\right],\end{array}$$E2

respectively. Thus, the space $W_2^3\left[0,1\right]$ is a reproducing kernel space, that is, for each fixed $y\in \left[0,1\right]$ and any $v\in W_2^3\left[0,1\right],$ there exists a function $R_y$ such that

$$v\left(y\right)={⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^3},$$E3

and similarly, we define the space

$$T_2^3\left[0,1\right]=\left\{\begin{array}{c}v\mid v,v^{\prime },{v^{\prime }}^{\prime }:\left[0,1\right]\to \mathbb{R}\mathrm{are}\text{absolutely\hspace{0.17em}continuous},\\ \\ {v^{\prime }}^{\prime }\in L^2\left[0,1\right],v\left(0\right)=0,v^{\prime }\left(0\right)=0\end{array}\right\}$$E4

The inner product and the norm in $T_2^3\left[0,1\right]$ are defined by

$$\begin{array}{c}{⟨v,ɡ⟩}_{T_2^3}=\sum _{i=0}^2{v}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\prime \prime \prime }\left(t\right){ɡ}^{\prime \prime \prime }\left(t\right)\mathrm{d}t,v,ɡ\in T_2^3\left[0,1\right],\\ \parallel v{\parallel }_{T_2^3}=\sqrt{{⟨v,v⟩}_{T_2^3}},v\in T_2^3\left[0,1\right],\end{array}$$E5

respectively. The space $T_2^3\left[0,1\right]$ is a reproducing kernel Hilbert space, and its reproducing kernel function $r_s$ is given by [1] as

$$r_s=\left\{\begin{array}{c}\frac{1}{4}{s}^2{t}^2+\frac{1}{12}{s}^2{t}^3-\frac{1}{24}{\mathit{st}}^4+\frac{1}{120}{t}^5,t\le s,\\ \\ \frac{1}{4}{s}^2{t}^2+\frac{1}{12}{s}^3{t}^2-\frac{1}{24}{\mathit{ts}}^4+\frac{1}{120}{s}^5,t>s,\end{array}\right.$$E6

and the space

$$G_2^1\left[0,1\right]=\left\{v|v:\left[0,1\right]\to \mathbb{R}\text{is\hspace{0.17em}absolutely\hspace{0.17em}continuous},v^{\prime }\left(x\right)\in L^2[0,1]\right\},$$E7

is a Hilbert space, where the inner product and the norm in $G_2^1\left[0,1\right]$ are defined by

$$\begin{array}{c}{⟨v,ɡ⟩}_{G_2^1}=v^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\prime }\left(x\right){ɡ}^{\prime }\left(x\right)\mathrm{d}x,v,ɡ\in G_2^1\left[0,1\right],\\ \parallel v{\parallel }_{G_2^1}=\sqrt{{⟨v,v⟩}_{G_2^1}},v\in G_2^1\left[0,1\right],\end{array}$$E8

respectively. The space $G_2^1\left[0,1\right]$ is a reproducing kernel space, and its reproducing kernel function $Q_y$ is given by [1] as

$$Q_y=\left\{\begin{array}{c}1+x,x⩽y\\ 1+y,x>y.\end{array}\right.$$E9

Theorem 1.1. The space $W_2^3\left[0,1\right]$ is a complete reproducing kernel space whose reproducing kernel $R_y$ is given by

$$R_y\left(x\right)=\left\{\begin{array}{l}\sum _{i=1}^6{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^6{d}_i\left(y\right)x^{i-1},x>y,\end{array}\right.$$E10

where

$$\begin{array}{c}{c}_1\left(y\right)=1,c_2\left(y\right)=y,c_3\left(y\right)=\frac{y^2}{4},c_4\left(y\right)=\frac{y^2}{12},c_5\left(y\right)=-\frac{1}{24y},c_6\left(y\right)=\frac{1}{120},\\ d_1\left(y\right)=1+\frac{y^5}{120},d_2\left(y\right)=\frac{-y^4}{24}+y,d_3\left(y\right)=\frac{y^2}{4}+\frac{y^3}{12},d_4\left(y\right)=d_5\left(y\right)=d_6\left(y\right)=0.\end{array}$$

Proof. Since

$${⟨v,R_y⟩}_{W_2^3}=\sum _{i=0}^2{v}^{\left(i\right)}\left(0\right)R_y^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\left(3\right)}\left(x\right)R_y^{\left(3\right)}\left(x\right)\mathrm{d}x,(v,R_y\in W_2^3\left[0,1\right]$$E11

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

$$\begin{array}{c}{⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^4}=\sum _{i=0}^2{v}^{\left(i\right)}\left(0\right)\left[R_y^{\left(i\right)}\left(0\right)-{\left(-1\right)}^{\left(2-i\right)}{R}_y^{\left(5-i\right)}\left(0\right)\right]\\ +\sum _{i=0}^2{\left(-1\right)}^{\left(2-i\right)}{v}^{\left(i\right)}\left(1\right)R_y^{\left(5-i\right)}\left(1\right)+{\int }_0^{1}v\left(x\right)R_y^{\left(6\right)}\left(x\right)\mathrm{d}x.\end{array}$$E12

Note, the property of the reproducing kernel as

$${⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^3}=v\left(y\right).$$E13

If

$$\begin{array}{c}{R}_y\left(0\right)-R_y^{\left(5\right)}\left(0\right)=0,\\ R_y^{\prime }\left(0\right)+R_y^{\left(4\right)}\left(0\right)=0,\\ R_y^{″}\left(0\right)-R_y^{‴}\left(0\right)=0,\\ R_y^{\left(3\right)}\left(1\right)=0,\\ R_y^{\left(4\right)}\left(1\right)=0,\\ R_y^{\left(5\right)}\left(1\right)=0,\end{array}$$E14

Then by (11), we obtain

$$R_y^{\left(6\right)}\left(x\right)=\delta \left(x-y\right),$$E15

when $x\ne y,$

$$R_y^{\left(6\right)}\left(x\right)=0,$$E16

therefore,

$$R_y\left(x\right)=\left\{\begin{array}{c}\begin{array}{l}\sum _{i=1}^6{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^6{d}_i\left(y\right)x^{i-1},x>y,\end{array}\end{array}\right.$$E17

Since

$$R_y^{\left(6\right)}\left(x\right)=\delta \left(x-y\right),$$E18

we have

$$\begin{array}{r}{\partial }^k{R}_{y^{+}}\left(y\right)={\partial }^k{R}_{y^{-}}\left(y\right),k=\mathrm{0,1,2,3,4},\\ {\partial }^5{R}_{y^{+}}\left(y\right)-{\partial }^5{R}_{y^{-}}\left(y\right)=-1.\end{array}$$E19

From (14) and (19), the unknown coefficients $c_i\left(y\right)$ and $d_i\left(y\right)$ $\left(i=1,2,\dots ,6\right)$ can be obtained. Thus, $R_y$ is given by

$$R_y=\left\{\begin{array}{c}\begin{array}{l}1+\mathit{yx}+\frac{1}{4}{y}^2{x}^2+\frac{1}{12}{y}^2{x}^3-\frac{1}{24}{\mathit{yx}}^4+\frac{1}{120}{x}^5,x\le y\\ 1+\mathit{yx}+\frac{1}{4}{y}^2{x}^2+\frac{1}{12}{y}^3{x}^2-\frac{1}{24}{\mathit{xy}}^4+\frac{1}{120}{y}^5,x>y.\end{array}\end{array}\right.$$E20

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

$$W\left(\mathrm{\Omega }\right)=\left\{\begin{array}{c}v\left(x,t\right)\mid \frac{{\partial }^{4}v}{\partial x^2\partial t^2},\text{is\hspace{0.17em}completely\hspace{0.17em}continuous\hspace{0.17em}in}\mathrm{\Omega }=\left[0,1\right]\times \left[0,1\right],\\ \\ \frac{{\partial }^{6}v}{\partial x^3\partial t^3}\in L^2\left(\mathrm{\Omega }\right),v\left(x,0\right)=0,\frac{\partial v\left(x,0\right)}{\partial t}=0\end{array}\right\}$$E21

is a binary reproducing kernel Hilbert space. The inner product and the norm in $W\left(\mathrm{\Omega }\right)$ are defined by

$$\begin{array}{cc}{⟨v\left(x,t\right),ɡ\left(x,t\right)⟩}_W& =\sum _{i=0}^2{\int }_0^1\left[\frac{{\partial }^3}{\partial t^3}\frac{{\partial }^i}{\partial x^i}v\left(0,t\right)\frac{{\partial }^3}{\partial t^3}\frac{{\partial }^i}{\partial x^i}ɡ(0,t)\right]\mathrm{d}t\\ & +\sum _{j=0}^2{⟨\frac{{\partial }^j}{\partial t^j}v\left(x,0\right),\frac{{\partial }^j}{\partial t^j}ɡ(x,0)⟩}_{W_2^3}\\ & +{\int }_0^1{\int }_0^1\left[\frac{{\partial }^3}{\partial x^3}\frac{{\partial }^3}{\partial t^3}v\left(x,t\right)\frac{{\partial }^3}{\partial x^3}\frac{{\partial }^3}{\partial t^3}ɡ(x,t)\right]\mathrm{d}x\mathrm{d}t,\\ & \parallel v{\parallel }_w=\sqrt{{⟨v,v⟩}_W},v\in W\left(\mathrm{\Omega }\right),\end{array}$$E22

respectively.

Theorem 1.2. The $W\left(\mathrm{\Omega }\right)$ is a reproducing kernel space, and its reproducing kernel function is

$$K_{\left(y,s\right)}=R_y{r}_s$$E23

such that for any $v\in W\left(\mathrm{\Omega }\right),$

$$\begin{array}{r}v\left(y,s\right)={⟨v\left(x,t\right),K_{\left(y,s\right)}\left(x,t\right)⟩}_{W^{\prime }}\\ K_{\left(y,s\right)}\left(x,t\right)=K_{\left(x,t\right)}\left(y,s\right).\end{array}$$E24

Similarly, the space

$$\widehat{W}\left(\mathrm{\Omega }\right)=\left\{v\left(x,t\right)|v(x,t)\mathit{\text{is\hspace{0.17em}completely\hspace{0.17em}continuous\hspace{0.17em}in}}\mathrm{\Omega }=[0,1]\times [0,1],\frac{{\partial }^{2}v}{\partial x\partial t}\in L^2\left(\mathrm{\Omega }\right)\right\}$$E25

is a binary reproducing kernel Hilbert space. The inner product and the norm in $\widehat{W}\left(\mathrm{\Omega }\right)$ are defined by [1] as

$$\begin{array}{cc}{⟨v\left(x,t\right),ɡ\left(x,t\right)⟩}_{{\displaystyle \widehat{W}}}& ={\int }_0^1\left[\frac{\partial }{\partial t}v\left(0,t\right)\frac{\partial }{\partial t}ɡ(0,t)\right]\mathrm{d}t+{⟨v\left(x,0\right),ɡ\left(x,0\right)⟩}_{W_2^1}\\ & +{\int }_0^1{\int }_0^1\left[\frac{\partial }{\partial x}\frac{\partial }{\partial t}v\left(x,t\right)\frac{\partial }{\partial x}\frac{\partial }{\partial t}ɡ(x,t)\right]\mathrm{d}x\mathrm{d}t,\\ & \parallel v{\parallel }_{{\displaystyle \widehat{W}}}=\sqrt{{⟨v,v⟩}_{{\displaystyle \widehat{W}}}},v\in {\displaystyle \widehat{W}}\left(\mathrm{\Omega }\right),\end{array}$$E26

respectively. $\widehat{W}\left(\mathrm{\Omega }\right)$ is a reproducing kernel space, and its reproducing kernel function $G_{\left(y,s\right)}$ is

$$G_{\left(y,s\right)}=Q_y{Q}_s.$$E27

Definition 1.3.

$$W_2^3\left[0,1\right]=\left\{\begin{array}{c}u\left(x\right)\mid u\left(x\right),u^{\prime }\left(x\right),{u^{\prime }}^{\prime }\left(x\right),\mathrm{are}\text{absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,1\right]\\ \\ u^{\left(3\right)}\left(x\right)\in L^2\left[0,1\right],x\in \left[0,1\right],u\left(0\right)=0,u\left(1\right)=0.\end{array}\right\}$$

The inner product and the norm in $W_2^3\left[0,1\right]$ are defined, respectively, by

$${⟨u\left(x\right),ɡ\left(x\right)⟩}_{W_2^3}=\sum _{i=0}^2{u}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^1{u}^{\left(3\right)}\left(x\right){ɡ}^{\left(3\right)}\left(x\right)\mathrm{d}x,u\left(x\right),ɡ\left(x\right)\in W_2^3\left[0,1\right]$$

and

$$\parallel u{\parallel }_{W_2^3}=\sqrt{{⟨u,u⟩}_{W_2^3}},u\in W_2^3\left[0,1\right].$$

The space $W_2^3\left[0,1\right]$ is a reproducing kernel space, that is, for each fixed $y\in \left[0,1\right]$ and any $u\left(x\right)\in W_2^3\left[0,1\right],$ there exists a function $R_y\left(x\right)$ such that

$$u\left(y\right)={⟨u\left(x\right),R_y\left(x\right)⟩}_{W_2^3}.$$

Definition 1.4.

$$W_2^1\left[0,1\right]=\left\{\begin{array}{c}u\left(x\right)\mid u\left(x\right),\text{is\hspace{0.17em}absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,1\right]\\ \\ u^{\prime }\left(x\right)\in L^2\left[0,1\right],x\in \left[0,1\right],\end{array}\right\}$$

The inner product and the norm in $W_2^1\left[0,1\right]$ are defined, respectively, by

$${⟨u\left(x\right),ɡ\left(x\right)⟩}_{W_2^1}=u\left(0\right)ɡ\left(0\right)+{\int }_0^1{u}^{\prime }\left(x\right){ɡ}^{\prime }\left(x\right)\mathrm{d}x,u\left(x\right),ɡ\left(x\right)\in W_2^1\left[0,1\right],$$E28

and

$$\parallel u{\parallel }_{W_2^1}=\sqrt{{⟨u,u⟩}_{W_2^1}},u\in W_2^1\left[0,1\right].$$E29

The space $W_2^1\left[0,1\right]$ is a reproducing kernel space, and its reproducing kernel function $T_x\left(y\right)$ is given by

$$T_x\left(y\right)=\left\{\begin{array}{c}1+x,x\le y,\\ 1+y,x>y.\end{array}\right.$$E30

Theorem 1.5. The space $W_2^3\left[0,1\right]$ is a complete reproducing kernel space, and its reproducing kernel function $R_y\left(x\right)$ can be denoted by

$$R_y\left(x\right)=\left\{\begin{array}{c}\begin{array}{l}\sum _{i=1}^6{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^6{d}_i\left(y\right)x^{i-1},x>y,\end{array}\end{array}\right.$$

where

$$\begin{array}{l}{c}_1\left(y\right)=0,\\ c_2\left(y\right)=\frac{5}{516}{y}^4-\frac{1}{156}{y}^5-\frac{5}{26}{y}^2-\frac{5}{78}{y}^3+\frac{3}{13}y,\\ c_3\left(y\right)=\frac{5}{624}{y}^4-\frac{1}{624}{y}^5+\frac{21}{104}{y}^2-\frac{5}{312}{y}^3-\frac{5}{26}y,\\ c_4\left(y\right)=\frac{5}{1872}{y}^4-\frac{1}{1872}{y}^5+\frac{7}{104}{y}^2-\frac{5}{936}{y}^3-\frac{5}{78}y,\\ c_5\left(y\right)=-\frac{5}{3744}{y}^4+\frac{1}{3744}{y}^5+\frac{5}{624}{y}^2+\frac{5}{1872}{y}^3-\frac{1}{104}y,\\ c_6\left(y\right)=\frac{1}{120}+\frac{1}{3744}{y}^4-\frac{1}{18720}{y}^5-\frac{1}{624}{y}^2-\frac{1}{1872}{y}^3-\frac{1}{156}y,\\ d_1\left(y\right)=\frac{1}{120}{y}^5,\\ d_2\left(y\right)=-\frac{1}{104}{y}^4-\frac{1}{156}{y}^5-\frac{5}{26}{y}^2-\frac{5}{78}{y}^3+\frac{3}{13}y,\\ d_3\left(y\right)=\frac{5}{624}{y}^4-\frac{1}{624}{y}^5+\frac{21}{104}{y}^2+\frac{7}{104}{y}^3-\frac{5}{26}y,\\ d_4\left(y\right)=\frac{5}{1872}{y}^4-\frac{1}{1872}{y}^5-\frac{5}{312}{y}^2-\frac{5}{936}{y}^3-\frac{5}{78}y,\\ d_5\left(y\right)=-\frac{5}{3744}{y}^4+\frac{1}{3744}{y}^5+\frac{5}{624}{y}^2+\frac{5}{1872}{y}^3+\frac{5}{156}y,\\ d_6\left(y\right)=-\frac{1}{156}y+\frac{1}{3744}{y}^4-\frac{1}{18720}{y}^5-\frac{1}{624}{y}^2-\frac{1}{1872}{y}^3.\end{array}$$

Proof. We have

$$\begin{array}{ll}{⟨u\left(x\right),R_y\left(x\right)⟩}_{W_2^3}=& \sum _{i=0}^2{u}^{\left(i\right)}\left(0\right)R_y^{\left(i\right)}\left(0\right)\\ & +{\int }_0^1{u}^{\left(3\right)}\left(x\right)R_y^{\left(3\right)}\left(x\right)\mathrm{d}x.\end{array}$$E31

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

$$\begin{array}{ll}{⟨u\left(x\right),R_y\left(x\right)⟩}_{W_2^6}=& \sum _{i=0}^2{u}^{\left(i\right)}\left(0\right)\left[R_y^{\left(i\right)}\left(0\right)-{\left(-1\right)}^{\left(2-i\right)}{R}_y^{\left(5-i\right)}\left(0\right)\right]\\ & +\sum _{i=0}^2{\left(-1\right)}^{\left(2-i\right)}{u}^{\left(i\right)}\left(1\right)R_y^{\left(5-i\right)}\left(1\right)\\ & -{\int }_0^{1}u\left(x\right)R_y^{\left(6\right)}\left(x\right)\mathrm{d}x.\end{array}$$E32

Note that property of the reproducing kernel

$${⟨u\left(x\right),R_y\left(x\right)⟩}_{W_2^3}=u\left(y\right),$$

If

$$\left\{\begin{array}{l}{R}_y^{{\text{'}}^{\prime }}\left(0\right)-R_y^{\left(3\right)}\left(0\right)=0,\\ R_y^{\text{'}}\left(0\right)+R_y^{\left(4\right)}\left(0\right)=0,\\ R_y^{\left(3\right)}\left(1\right)=0,\\ R_y^{\left(4\right)}\left(1\right)=0,\end{array}\right.$$E33

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

$$-R_y^{\left(6\right)}\left(x\right)=\delta \left(x-y\right),$$

$$\text{when} x\ne y,$$

$$R_y^{\left(6\right)}\left(x\right)=0,$$

therefore,

$$R_y\left(x\right)=\left\{\begin{array}{c}\begin{array}{l}\sum _{i=1}^6{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^6{d}_i\left(y\right)x^{i-1},x>y,\end{array}\end{array}\right.$$

Since

$$-R_y^{\left(6\right)}\left(x\right)=\delta \left(x-y\right),$$

we have

$${\partial }^k{R}_{y^{+}}\left(y\right)={\partial }^k{R}_{y^{-}}\left(y\right),k=\mathrm{0,1,2,3,4},$$E34

and

$${\partial }^5{R}_{y^{+}}\left(y\right)-{\partial }^5{R}_{y^{-}}\left(y\right)=-1.$$E35

Since $R_y\left(x\right)\in W_2^3\left[0,1\right],$ it follows that

$$R_y\left(0\right)=0,R_y\left(1\right)=0,$$E36

From (33)–(36), the unknown coefficients $c_i\left(y\right)$ and $d_i\left(y\right)$ $\left(i=1,2,\dots ,6\right)$ can be obtained. Thus $R_y\left(x\right)$ is given by

$$R_y\left(x\right)=\left\{\begin{array}{l}\frac{5}{516}{\mathit{xy}}^4-\frac{1}{156}{\mathit{xy}}^5-\frac{5}{26}{\mathit{xy}}^2-\frac{5}{78}{\mathit{xy}}^3+\frac{3}{13}\mathit{xy}+\frac{5}{624}{x}^2{y}^4-\frac{1}{624}{x}^2{y}^5+\frac{21}{104}{x}^2{y}^2\\ -\frac{5}{312}{x}^2{y}^3-\frac{5}{26}{x}^{2}y+\frac{5}{1872}{x}^3{y}^4-\frac{1}{1872}{x}^3{y}^5+\frac{7}{104}{x}^3{y}^2-\frac{5}{936}{x}^3{y}^3-\frac{5}{78}{x}^{3}y\\ -\frac{5}{3744}{x}^4{y}^4+\frac{1}{3744}{x}^4{y}^5+\frac{5}{624}{x}^4{y}^2+\frac{5}{1872}{x}^4{y}^3-\frac{1}{104}{x}^{4}y-\frac{1}{156}{x}^{5}y+\frac{1}{3744}{x}^5{y}^4\\ -\frac{1}{18720}{x}^5{y}^5-\frac{1}{624}{x}^5{y}^2-\frac{1}{1872}{x}^5{y}^3,x\le y\\ \frac{5}{516}{\mathit{yx}}^4-\frac{1}{156}{\mathit{yx}}^5-\frac{5}{26}{\mathit{yx}}^2-\frac{5}{78}{\mathit{yx}}^3+\frac{3}{13}\mathit{xy}+\frac{5}{624}{y}^2{x}^4-\frac{1}{624}{y}^2{x}^5+\frac{21}{104}{x}^2{y}^2\\ -\frac{5}{312}{y}^2{x}^3-\frac{5}{26}{y}^{2}x+\frac{5}{1872}{y}^3{x}^4-\frac{1}{1872}{y}^3{x}^5+\frac{7}{104}{y}^3{x}^2-\frac{5}{936}{x}^3{y}^3-\frac{5}{78}{y}^{3}x\\ -\frac{5}{3744}{x}^4{y}^4+\frac{1}{3744}{y}^4{x}^5+\frac{5}{624}{y}^4{x}^2+\frac{5}{1872}{y}^4{x}^3-\frac{1}{104}{y}^{4}x-\frac{1}{156}{y}^{5}x+\frac{1}{3744}{y}^5{x}^4\\ -\frac{1}{18720}{x}^5{y}^5-\frac{1}{624}{y}^5{x}^2-\frac{1}{1872}{y}^5{x}^3,x>y\end{array}\right.$$

$$W_2^4\left[0,1\right]=\left\{\begin{array}{c}v\left(x\right)\mid v\left(x\right),v^{\prime }\left(x\right),{v^{\prime \prime }}^{}\left(x\right),{{v^{\prime \prime \prime }}^{}}^{}\left(x\right)\\ \\ \mathrm{are}\text{absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,1\right],\\ \\ v^{\left(4\right)}\left(x\right)\in L^2\left[0,1\right],x\in \left[0,1\right]\end{array}\right\}$$E37

The inner product and the norm in $W_2^4\left[0,1\right]$ are defined, respectively, by

$$\begin{array}{c}{⟨v\left(x\right),ɡ\left(x\right)⟩}_{W_2^4}=\sum _{i=0}^3{v}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\left(4\right)}\left(x\right){ɡ}^{\left(4\right)}\left(x\right)\mathrm{d}x,v\left(x\right),ɡ\left(x\right)\in W_2^4\left[0,1\right],\\ \parallel v{\parallel }_{W_2^4}=\sqrt{{⟨v,v⟩}_{W_2^4}},v\in W_2^4\left[0,1\right].\end{array}$$E38

The space $W_2^4\left[0,1\right]$ is a reproducing kernel space, that is, for each fixed.

$y\in \left[0,1\right]$ and any $v\left(x\right)\in W_2^4\left[0,1\right],$ there exists a function $R_y\left(x\right)$ such that

$$v\left(y\right)={⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^4}$$E39

Similarly, we define the space

$$W_2^2\left[0,T\right]=\left\{\begin{array}{c}v\left(t\right)\mid v\left(t\right),v^{\prime }\left(t\right)\\ \\ \mathrm{are}\text{absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,T\right],\\ \\ {v^{\prime }}^{\prime }\left(t\right)\in L^2\left[0,T\right],t\in \left[0,T\right],v\left(0\right)=0\\ \end{array}\right\}$$E40

The inner product and the norm in $W_2^2\left[0,T\right]$ are defined, respectively, by

$$\begin{array}{c}{⟨v\left(t\right),ɡ\left(t\right)⟩}_{W_2^2}=\sum _{i=0}^1{v}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^T{v^{\prime \prime }}^{}\left(t\right){{ɡ}^{\prime \prime }}^{}\left(t\right)\mathrm{d}t,v\left(t\right),ɡ\left(t\right)\in W_2^2\left[0,T\right],\\ \parallel v{\parallel }_{W_1}=\sqrt{{⟨v,v⟩}_{W_2^2}},v\in W_2^2\left[0,T\right].\end{array}$$E41

Thus, the space $W_2^2\left[0,T\right]$ is also a reproducing kernel space, and its reproducing kernel function $r_s\left(t\right)$ can be given by

$$r_s\left(t\right)=\left\{\begin{array}{c}\mathit{st}+\frac{s}{2}{t}^2-\frac{1}{6}{t}^3,t\le s,\\ \mathit{st}+\frac{t}{2}{s}^2-\frac{1}{6}{s}^3,t>s,\end{array}\right.$$E42

and the space

$$W_2^2\left[0,1\right]=\left\{\begin{array}{c}v\left(x\right)\mid v\left(x\right),v^{\prime }\left(x\right)\\ \\ \mathrm{are}\text{absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,1\right],\\ \\ {v^{\prime \prime }}^{}\left(x\right)\in L^2\left[0,1\right],x\in \left[0,1\right]\\ \end{array}\right\}$$E43

where the inner product and the norm in $W_2^2\left[0,1\right]$ are defined, respectively, by

$$\begin{array}{c}{⟨v\left(t\right),ɡ\left(t\right)⟩}_{W_2^2}=\sum _{i=0}^1{v}^{\left(i\right)}\left(0\right){ɡ}^{\left(i\right)}\left(0\right)+{\int }_0^T{v^{\prime \prime }}^{}\left(t\right){{ɡ}^{\prime \prime }}^{}\left(t\right)\mathrm{d}t,v\left(t\right),ɡ\left(t\right)\in W_2^2\left[0,1\right],\\ \parallel v{\parallel }_{W_2}=\sqrt{{⟨v,v⟩}_{W_2^2}},v\in W_2^2\left[0,1\right].\end{array}$$E44

The space $W_2^2\left[0,1\right]$ is a reproducing kernel space, and its reproducing kernel function $Q_y\left(x\right)$ is given by

$$Q_y\left(x\right)=\left\{\begin{array}{c}1+\mathit{xy}+\frac{y}{2}{x}^2-\frac{1}{6}{x}^3,x\le y,\\ 1+\mathit{xy}+\frac{x}{2}{y}^2-\frac{1}{6}{y}^3,x>y.\end{array}\right.$$E45

Similarly, the space $W_2^1\left[0,T\right]$ is defined by

$$W_2^1\left[0,T\right]=\left\{\begin{array}{c}v\left(t\right)\mid v\left(t\right)\text{is\hspace{0.17em}absolutely\hspace{0.17em}continuous\hspace{0.17em}in}\left[0,T\right],\\ \\ v\left(t\right)\in L^2\left[0,T\right],t\in \left[0,T\right]\end{array}\right\}$$E46

The inner product and the norm in $W_2^1\left[0,T\right]$ are defined, respectively, by

$$\begin{array}{c}{⟨v\left(t\right),ɡ\left(t\right)⟩}_{W_2^1}=v\left(0\right)ɡ\left(0\right)+{\int }_0^T{v}^{\prime }\left(t\right){ɡ}^{\prime }\left(t\right)\mathrm{d}t,v\left(t\right),ɡ\left(t\right)\in W_2^1\left[0,T\right],\\ \parallel v{\parallel }_{W_2^1}=\sqrt{{⟨v,v⟩}_{W_2^1}},v\in W_2^1\left[0,T\right].\end{array}$$E47

The space $W_2^1\left[0,T\right]$ is a reproducing kernel space, and its reproducing kernel function $q_s\left(t\right)$ is given by

$$q_s\left(t\right)=\left\{\begin{array}{c}1+t,t\le s,\\ 1+s,t>s.\end{array}\right.$$E48

Further, we define the space $W\left(\mathrm{\Omega }\right)$ as

$$W\left(\mathrm{\Omega }\right)=\left\{\begin{array}{c}v\left(x,t\right)\mid \frac{{\partial }^{4}v}{\partial x^3\partial t},\text{is\hspace{0.17em}completely\hspace{0.17em}continuous},\\ \\ \mathit{\text{in}}\mathrm{\Omega }=\left[0,1\right]\times \left[0,T\right],\\ \\ \frac{{\partial }^{6}v}{\partial x^4\partial t^2}\in L^2\left(\mathrm{\Omega }\right),v\left(x,0\right)=0\\ \end{array}\right\}$$E49

and the inner product and the norm in $W\left(\mathrm{\Omega }\right)$ are defined, respectively, by

$$\begin{array}{ll}{⟨v\left(x,t\right),ɡ\left(x,t\right)⟩}_W& =\sum _{i=0}^3{\int }_0^T\left[\frac{{\partial }^2}{\partial t^2}\frac{{\partial }^i}{\partial x^i}v\left(0,t\right)\frac{{\partial }^2}{\partial t^2}\frac{{\partial }^i}{\partial x^i}ɡ(0,t)\right]\mathrm{d}t\\ & +\sum _{j=0}^1{⟨\frac{{\partial }^j}{\partial t^j}v\left(x,0\right),\frac{{\partial }^j}{\partial t^j}ɡ(x,0)⟩}_{W_2^4}\\ & +{\int }_0^T{\int }_0^1\left[\frac{{\partial }^4}{\partial x^4}\frac{{\partial }^2}{\partial t^2}v\left(x,t\right)\frac{{\partial }^4}{\partial x^4}\frac{{\partial }^2}{\partial t^2}ɡ(x,t)\right]\mathrm{d}x\mathrm{d}t,\\ & \parallel v{\parallel }_W=\sqrt{{⟨v,v⟩}_W},v\in W\left(\mathrm{\Omega }\right).\end{array}$$E50

Now, we have the following theorem:

Theorem 1.6. The space $W_2^4\left[0,1\right]$ is a complete reproducing kernel space, and its reproducing kernel function $R_y\left(x\right)$ can be denoted by

$$R_y\left(x\right)=\left\{\begin{array}{c}\begin{array}{l}\sum _{i=1}^8{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^8{d}_i\left(y\right)x^{i-1},x>y,\end{array}\end{array}\right.$$E51

where

$$\begin{array}{ccc}{c}_1\left(y\right)=1,& c_2\left(y\right)=y,& c_3\left(y\right)=\frac{1}{4}{y}^2,\\ c_4\left(y\right)=\frac{1}{36}{y}^3,& c_5\left(y\right)=\frac{1}{144}{y}^3,& c_6\left(y\right)=-\frac{1}{240}{y}^2,\\ & c_7\left(y\right)=\frac{1}{720}y,& c_8\left(y\right)=-\frac{1}{5040},\\ & d_1\left(y\right)=1-\frac{1}{5040}{y}^7,& d_2\left(y\right)=y+\frac{1}{720}{y}^6,\\ & d_3\left(y\right)=\frac{1}{4}{y}^2-\frac{1}{240}{y}^5,& d_4\left(y\right)=\frac{1}{36}{y}^3+\frac{1}{144}{y}^4,\\ d_5\left(y\right)=0,& d_6\left(y\right)=0,& d_7\left(y\right)=0,d_8\left(y\right)=0.\end{array}$$E52

Proof. Since

$$\begin{array}{r}{⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^4}=\sum _{i=0}^3{v}^{\left(i\right)}\left(0\right)R_y^{\left(i\right)}\left(0\right)+{\int }_0^1{v}^{\left(4\right)}\left(x\right)R_y^{\left(4\right)}\left(x\right)\mathrm{d}x,\\ \left(v\left(x\right),R_y\left(x\right)\in W_2^4\left[0,1\right]\right)\end{array}$$E53

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

$$\begin{array}{ll}{⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^4}=& \sum _{i=0}^3{v}^{\left(i\right)}\left(0\right)\left[R_y^{\left(i\right)}\left(0\right)-{\left(-1\right)}^{\left(3-i\right)}{R}_y^{\left(7-i\right)}\left(0\right)\right]\\ & +\sum _{i=0}^3{\left(-1\right)}^{\left(3-i\right)}{v}^{\left(i\right)}\left(1\right)R_y^{\left(7-i\right)}\left(1\right)\\ & +{\int }_0^{1}v\left(x\right)R_y^{\left(8\right)}\left(x\right)\mathrm{d}x.\end{array}$$E54

Note that property of the reproducing kernel

$${⟨v\left(x\right),R_y\left(x\right)⟩}_{W_2^4}=v\left(y\right).$$E55

If

$$\begin{array}{r}{R}_y\left(0\right)+R_y^{\left(7\right)}\left(0\right)=0,\\ R_y^{\prime }\left(0\right)-R_y^{\left(6\right)}\left(0\right)=0,\\ R_y^{″}\left(0\right)+R_y^{\left(5\right)}\left(0\right)=0,\\ R_y^{‴}\left(0\right)-R_y^{\left(4\right)}\left(0\right)=0,\\ R_y^{\left(4\right)}\left(1\right)=0,\\ R_y^{\left(5\right)}\left(1\right)=0,\\ R_y^{\left(6\right)}\left(1\right)=0,\\ R_y^{\left(7\right)}\left(1\right)=0,\end{array}$$E56

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

$$R_y^{\left(8\right)}\left(x\right)=\delta \left(x-y\right),$$E57

when $x\ne y,$

$$R_y^{\left(8\right)}\left(x\right)=0;$$E58

therefore,

$$R_y\left(x\right)=\left\{\begin{array}{c}\begin{array}{l}\sum _{i=1}^8{c}_i\left(y\right)x^{i-1},x\le y,\\ \sum _{i=1}^8{d}_i\left(y\right)x^{i-1},x>y.\end{array}\end{array}\right.$$E59

Since

$$R_y^{\left(8\right)}\left(x\right)=\delta \left(x-y\right),$$E60

we have

$${\partial }^k{R}_{y^{+}}\left(y\right)={\partial }^k{R}_{y^{-}}\left(y\right),k=\mathrm{0,1,2,3,4,5,6},$$E61

$${\partial }^7{R}_{y^{+}}\left(y\right)-{\partial }^7{R}_{y^{-}}\left(y\right)=1.$$E62

From (56)–(62), the unknown coefficients $c_i\left(y\right)$ ve $d_i\left(y\right)\left(i=1,2,\dots ,8\right)$ can be obtained. Thus, $R_y\left(x\right)$ is given by

$$R_y\left(x\right)=\left(\begin{array}{c}1+\mathit{yx}+\frac{1}{4}{y}^2{x}^2+\frac{1}{36}{y}^3{x}^3+\frac{1}{144}{y}^3{x}^4\\ \\ -\frac{1}{240}{y}^2{x}^5+\frac{1}{720}{\mathit{yx}}^6-\frac{1}{5040}{x}^7,x\le y,\\ \\ 1+\mathit{xy}+\frac{1}{4}{x}^2{y}^2+\frac{1}{36}{x}^3{y}^3+\frac{1}{144}{x}^3{y}^4\\ \\ -\frac{1}{240}{x}^2{y}^5+\frac{1}{720}{\mathit{xy}}^6-\frac{1}{5040}{y}^7,x>y.\end{array}\right.$$E63