Reproducing Kernel Hilbert Spaces of Smooth Fractal Interpolation Functions

Abstract

The theory of reproducing kernel Hilbert spaces (RKHSs) has been developed into a powerful tool in mathematics and has lots of applications in many fields, especially in kernel machine learning. Fractal theory provides new technologies for making complicated curves and fitting experimental data. Recently, combinations of fractal interpolation functions (FIFs) and methods of curve estimations have attracted the attention of researchers. We are interested in the study of connections between FIFs and RKHSs. The aim is to develop the concept of smooth fractal-type reproducing kernels and RKHSs of smooth FIFs. In this paper, a linear space of smooth FIFs is considered. A condition for a given finite set of smooth FIFs to be linearly independent is established. For such a given set, we build a fractal-type positive semi-definite kernel and show that the span of these linearly independent smooth FIFs is the corresponding RKHS. The nth derivatives of these FIFs are investigated, and properties of related positive semi-definite kernels and the corresponding RKHS are studied. We also introduce subspaces of these RKHS which are important in curve-fitting applications.

1. Introduction

The basic theory of reproducing kernel Hilbert spaces (RKHSs) was studied by Aronszajn [1], and it has been developed to be a powerful tool in operator theory, differential equations, integral equations, probability, statistics, and learning theory. See the excellent monographs [2,3,4,5,6,7] and references therein. Nowadays, learning algorithms in RKHSs play an important role in the development of machine learning. A linear model that can only be formulated as an inner product can be transformed into a nonlinear model by replacing the inner product with a symmetric positive semi-definite kernel. We use a kernel to map input data implicitly into a high-dimensional feature space to improve the performance of a learning algorithm. This is a widely used principle called the “kernel trick”. In many applications, data are arising from unknown functions, and it is required to generate a function to interpolate or approximate these data. The concept of RKHSs has been widely applied to such regression problems.

Various types of functions, such as polynomials, splines, rational functions, trigonometric functions, and wavelets have been widely-used in real-world data-fitting. However, sampled signals may have irregular forms in many practical problems, and fractal theory provides new technologies for making complicated curves and fitting experimental data. The theory of fractal interpolation functions (FIFs) is developed for the interpolation problem with a class of fractal functions. It generalizes traditional interpolation techniques through the property of self-similarity. The concept of FIFs defined through an iterated function system was introduced by Barnsley [8,9]. See also these books [10,11,12] for developments of the theory of FIFs and their applications.

Smooth FIFs have also been discussed by many authors. A construction of ${\mathcal{C}}^p$-FIFs was given in [13]. Based on this work, ${\mathcal{C}}^p$-Hermite FIFs were obtained in [14,15], ${\mathcal{C}}^2$-cubic spline FIFs were discussed in [16], and smooth rational cubic FIFs were investigated in [17,18,19,20]. Error bounds, shape-preserving properties, and parameter identification of smooth FIFs have been extensively discussed in the literature.

In [21,22,23], linear FIFs and recurrent linear FIFs were applied to model discrete sequences. In [24], estimations in RKHSs with dependent data were investigated. Recently, the combination of FIFs and other types of curve estimations has attracted the attention of researchers. In [25], the authors studied a fractal perturbation of a type of nonparametric curve estimation. In [26], a training set of samples was used to train an SVM model, and then a linear FIF was constructed based on the predicted data of SVM. In [27], fractal-type reproducing kernels and RKHSs of fractal functions were established. In [28], the author showed that a set of FIFs is an RKHS under two different types of inner products, and then apply such RKHSs to curve-fitting problems. Through the work given in [27,28], connections between FIFs and RKHSs are clearer and we see a new direction of research in the theory of FIFs and RKHSs. Since smooth FIFs have been studied by many researchers, it is natural to develop the concept of RKHSs of smooth FIFs. The purpose of this paper is to construct smooth fractal-type reproducing kernels and RKHSs of smooth FIFs.

Throughout this paper, let $t_0,t_1,\cdots ,t_N$ be a given set of real numbers such that $t_0<t_1<t_2<\cdots <t_N$, where $N$ is a positive integer and $N\ge 2$, and let $I=\left[t_0, t_N\right]$, For each $i=1, \cdots , N$, let $I_i=\left[t_{i-1}, t_i\right]$. We will denote by $\mathcal{C}\left[I\right]$ the set of all real-valued continuous functions defined on $I$. Define ${\Vert f\Vert }_{\infty }={\max }_{t\in I}\left|f\left(t\right)\right|$ for $f\in \mathcal{C}\left[I\right]$. For a nonnegative integer $p$, let ${\mathcal{C}}^p\left[I\right]$ denote the space of all real-valued functions whose $p$th derivatives exist and are continuous on $I$. We also denote ${\mathcal{C}}^p\left[I\right]$ by $\mathcal{C}\left[I\right]$ in the case $p=0$.

This paper is structured as follows. In Section 2, we give a brief introduction to the construction of smooth FIFs by the approach given in [13]. For given numbers $t_0, t_1, \cdots , t_N$ and a nonnegative integer $p$, a class of FIFs in ${\mathcal{C}}^p\left[I\right]$ is established. In Section 3, suppose that $t_0, \cdots , t_N$ and $p$ are fixed, and all vertical scaling factors in the construction of FIFs are also fixed numbers. We consider a linear space of smooth FIFs since linear combinations of smooth FIFs are also smooth FIFs. A condition for a set of smooth FIFs $\mathbb{B}=\left\{{\psi }_1, \cdots , {\psi }_m\right\}$ to be linearly independent is given. Then we establish a fractal-type positive semi-definite kernel $k$ by functions in $\mathbb{B}$, and show that ${\mathcal{F}}_{\mathbb{B}}=\mathrm{span}\left\{{\psi }_1, \cdots , {\psi }_m\right\}$ is a RKHS with the reproducing kernel $k$. A subspace $\mathcal{H}$ of ${\mathcal{F}}_{\mathbb{B}}$, which is important in curve fitting problems, is considered. To investigate the space of $n$th derivatives of functions in ${\mathcal{F}}_{\mathbb{B}}$, we consider the space ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}=\mathrm{span}\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}$ and show that ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$ is a RKHS with a reproducing kernel defined by ${\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}$, where ${\psi }_i^{\left(n\right)}$ is the $n$th derivative of ${\psi }_i$. Two subspaces of ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$, ${\mathcal{H}}^{\left(n\right)}$ and ${ℍ}^{\left(n\right)}$, are considered. We investigate connections between ${\mathcal{H}}^{\left(n\right)}$ and ${ℍ}^{\left(n\right)}$, and prove that if the ranks of $\left[{\psi }_l\left(t_i\right)\right]$ and $\left[{\psi }_l^{\left(n\right)}\left(t_i\right)\right]$ are both equal to $m$, then ${\mathcal{H}}^{\left(n\right)}={ℍ}^{\left(n\right)}$. Hence for any function $f$ in $\mathcal{H}$, we have two equivalent representations for the $n$th derivative of $f$.

2. Construction of Smooth Fractal Interpolation Functions

The construction of smooth FIFs given here has been treated in [13]. We show the details here to make our paper more self-contained.

Let $p$ be a nonnegative integer. For $i=1,\cdots ,N$, suppose $q_i\in {\mathcal{C}}^P\left[I\right]$, and define $L_i=I\to I_i$ and $M_i:I\times ℝ\to ℝ$ by

$$L_i\left(t\right)=\left(\frac{t_i-t_{i-1}}{t_N-t_0}\right)t+\left(\frac{t_N{t}_{i-1}-t_0{t}_i}{t_N-t_0}\right),$$

(1)

$$M_i\left(t,y\right)=s_{i}y+q_i\left(t\right).$$

(2)

Here $s_1,\dots ,s_N\in \left(-1,1\right)$, and these numbers are called vertical scaling factors. Assume that

$$\left|s_i\right|<a_i^p, \mathrm{where} a_i=\frac{t_i-t_{i-1}}{t_N-t_0}.$$

(3)

We also define

$$F_{i,n}\left(t,y\right)=\frac{s_{i}y+q_i^{\left(n\right)}\left(t\right)}{a_i^n}, y_{0,n}=\frac{q_1^{\left(n\right)}\left(t_0\right)}{a_1^n-s_1}, y_{N,n}=\frac{q_N^{\left(n\right)}\left(t_N\right)}{a_N^n-s_N},$$

(4)

for $i=1,\cdots ,N$, and $n=0, 1, \cdots , p$. Here $q_i^{\left(n\right)}$ is the $n$th derivative of $q_i$ and $q_i^{\left(0\right)}=q_i$.

In the case $n=0$, $F_{i, 0}=M_i$, and it is proved in ([8] Theorem 1) that if

$$F_{1, 0}\left(t_0, y_{0,0}\right)=y_{0,0}, F_{N, 0}\left(t_N, y_{N,0}\right)=y_{N,0},$$

and

$$F_{i-1, 0}\left(t_N, y_{N,0}\right)=F_{i, 0}\left(t_0, y_{0,0}\right) \mathrm{hold} \mathrm{for} i=2,\cdots ,N,$$

then $\left\{\left(L_i\left(t\right), M_i\left(t, y\right)\right):i=1, \cdots , N\right\}$ determines a FIF $f\in \mathcal{C}\left[I\right]$. We call $f$ a FIF corresponding to the set of scaling factors and functions $\left\{s_i;q_i: i=1, \cdots , N\right\}$. Moreover, the obtained FIF $f$ satisfies $f\left(t_0\right)=y_{0,0}, f\left(t_N\right)=y_{N,0}$, and

$$f\left(t_i\right)=y_i, \mathrm{where} y_i=s_i{y}_{N,0}+q_i\left(t_N\right), i=1, \cdots , N-1.$$

In the case $p\ge 1$, it is proved in ([13] Theorem 2) that if

$$F_{i-1,n}\left(t_N, y_{N,n}\right)=F_{i, n}\left(t_0, y_{0,n}\right), i=2, \cdots , N, n=1, \cdots , p,$$

(5)

then $\left\{\left(L_i\left(t\right), M_i\left(t, y\right)\right):i=1, \cdots , N\right\}$ determines a FIF $f\in {\mathcal{C}}^p\left[I\right]$, and $f^{\left(n\right)}$ is the FIF determined by $\left\{\left(L_i\left(t\right), F_{i,n}\left(t, y\right)\right):i=1, \cdots , N\right\}$ for $n=1, \cdots , p$. Moreover, for $n=0, 1, \cdots , p$,

$$f^{\left(n\right)}\left(t\right)=\frac{1}{a_i^n}\left\{s_i{f}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)+q_i^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)\right\}, t\in I_i, i=1, \cdots , N.$$

(6)

We see that $f^{\left(n\right)}$ is a FIF corresponding to the set $\left\{\frac{s_i}{a_i^n};\frac{q_i^{\left(n\right)}}{a_i^n}:i=1, \cdots , N\right\}.$

By $F_{i, 0}=M_i$ and Equation $\left(2\right)$, conditions $F_{1, 0}\left(t_0, y_{0,0}\right)=y_{0,0}$ and $F_{N, 0}\left(t_N, y_{N,0}\right)=y_{N,0}$ imply that

$$y_{0,0}=\frac{q_1\left(t_0\right)}{1-s_1} \mathrm{and} y_{N,0}=\frac{q_N\left(t_N\right)}{1-s_N},$$

which are consistent with that given in Equation $\left(4\right)$ with $n=0$. The conditions $F_{i-1, 0}\left(t_N, y_{N,0}\right)=F_{i, 0}\left(t_0, y_{0,0}\right)$ for $i=2,\cdots ,N$ can be reduced to

$$\frac{s_{i-1}}{1-s_N}{q}_N\left(t_N\right)+q_{i-1}\left(t_N\right)=\frac{s_i}{1-s_1}{q}_1\left(t_0\right)+q_i\left(t_0\right), i=2, \cdots , N.$$

(7)

By Equation$\left(4\right)$, condition Equation $\left(5\right)$ can be reduced to

$$\frac{s_{i-1}{q}_N^{\left(n\right)}\left(t_N\right)+\left(a_N^n-s_N\right)q_{i-1}^{\left(n\right)}\left(t_N\right)}{a_{i-1}^n\left(a_N^n-s_N\right)}=\frac{s_i{q}_1^{\left(n\right)}\left(t_0\right)+\left(a_1^n-s_1\right)q_i^{\left(n\right)}\left(t_0\right)}{a_i^n\left(a_1^n-s_1\right)}$$

(8)

for $i=2, \cdots , N$ and $n=1, \cdots , p$. Note that in the case $n=0$, Equation $\left(8\right)$ is reduced to Equation $\left(7\right)$.

3. Reproducing Kernel Hilbert Spaces of Smooth Fractal Interpolants

Throughout this section, suppose that $p$ is a fixed nonnegative integer, $N\ge 2$ is a fixed positive integer, and $t_0, \cdots , t_N$, $s_1, \cdots , s_N$ are all fixed numbers. Let $L_1, \cdots , L_N$ be given by Equation $\left(1\right)$.

3.1. Linear Spaces of FIFs and a Condition for Linearly Independent FIFs

Suppose that ${\mathcal{F}}_p$ is a set of functions in ${\mathcal{C}}^p\left[I\right]$ such that each $f$ in ${\mathcal{F}}_p$ is a ${\mathcal{C}}^p$-FIF corresponding to a set $\left\{s_i;q_i: i=1, \cdots , N\right\}$, where each $q_i$ is a function in ${\mathcal{C}}^p\left[I\right]$, and Equation $\left(8\right)$ holds for $i=2, \cdots , N$ and $n=0, 1, \cdots , p$. By the results given in Section 2 and Equation $\left(6\right)$, we see that $f^{\left(n\right)}$ is a FIF corresponding to the set $\left\{\frac{s_i}{a_i^n};\frac{q_i^{\left(n\right)}}{a_i^n}:i=1, \cdots , N\right\}$.

Proposition 1.

${\mathcal{F}}_p$ is a linear space.

Proof. 

It is easy to see that the zero function is in ${\mathcal{F}}_p$ with $q_i\equiv 0$ for each $i$. Suppose $f,g\in {\mathcal{F}}_p$. Then $f$ and $g$ are ${\mathcal{C}}^p$-FIFs corresponding to the sets $\left\{s_i;q_i: i=1, \cdots , N\right\}$ and $\left\{s_i;r_i: i=1, \cdots , N\right\}$, respectively, where $q_i,r_i\in {\mathcal{C}}^p\left[I\right]$ for $i=1, \cdots , N$, and satisfy Equation $\left(8\right)$. Therefore, for $n=0, 1, \cdots , p$ and $i=1, \cdots , N$,

$$f^{\left(n\right)}\left(t\right)=\frac{1}{a_i^n}\left\{s_i{f}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)+q_i^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)\right\}, t\in I_i,$$

(9)

$$g^{\left(n\right)}\left(t\right)=\frac{1}{a_i^n}\left\{s_i{g}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)+r_i^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)\right\}, t\in I_i,$$

(10)

and the following conditions hold for $i=2, \cdots , N$ and $n=0, 1, \cdots , p$:

$$\frac{s_{i-1}{q}_N^{\left(n\right)}\left(t_N\right)+\left(a_N^n-s_N\right)q_{i-1}^{\left(n\right)}\left(t_N\right)}{a_{i-1}^n\left(a_N^n-s_N\right)}=\frac{s_i{q}_1^{\left(n\right)}\left(t_0\right)+\left(a_1^n-s_1\right)q_i^{\left(n\right)}\left(t_0\right)}{a_i^n\left(a_1^n-s_1\right)},$$

(11)

$$\frac{s_{i-1}{r}_N^{\left(n\right)}\left(t_N\right)+\left(a_N^n-s_N\right)r_{i-1}^{\left(n\right)}\left(t_N\right)}{a_{i-1}^n\left(a_N^n-s_N\right)}=\frac{s_i{r}_1^{\left(n\right)}\left(t_0\right)+\left(a_1^n-s_1\right)r_i^{\left(n\right)}\left(t_0\right)}{a_i^n\left(a_1^n-s_1\right)}.$$

(12)

Let $a, b\in ℝ$. Then $af+bg$ satisfies

$${\left(af+bg\right)}^{\left(n\right)}\left(t\right)=\frac{1}{a_i^n}\left\{s_i{\left(af+bg\right)}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)+{\left(a{q}_i+b{r}_i\right)}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)\right\}, t\in I_i,$$

(13)

for $n=0, 1, \cdots , p$ and $i=1, \cdots , N$. Here $a{q}_i+b{r}_i\in {\mathcal{C}}^p\left[I\right]$ and satisfies

$$\begin{gathered} \frac{s_{i-1}{\left(a{q}_N+b{r}_N\right)}^{\left(n\right)}\left(t_N\right)+\left(a_N^n-s_N\right){\left(a{q}_{i-1}+b{r}_{i-1}\right)}^{\left(n\right)}\left(t_N\right)}{a_{i-1}^n\left(a_N^n-s_N\right)} \\ =\frac{s_i{\left(a{q}_1+b{r}_1\right)}^{\left(n\right)}\left(t_0\right)+\left(a_1^n-s_1\right){\left(a{q}_i+b{r}_i\right)}^{\left(n\right)}\left(t_0\right)}{a_i^n\left(a_1^n-s_1\right)} \end{gathered}$$

for $i=2, \cdots , N$ and $n=0, 1, \cdots , p$. Therefore $af+bg\in {\mathcal{F}}_p$ and is corresponding to the set $\left\{s_i;a{q}_i+b{r}_i: i=1, \cdots , N\right\}$. Moreover, $a{f}^{\left(n\right)}+b{g}^{\left(n\right)}$ is a FIF corresponding to the set $\left\{\frac{s_i}{a_i^n};\frac{a{q}_i^{\left(n\right)}+b{r}_i^{\left(n\right)}}{a_i^n}:i=1, \cdots , N\right\}$. □

When considering a subspace of ${\mathcal{F}}_p$, we are usually interested in a set of linearly independent functions in ${\mathcal{F}}_p$. Each FIF in ${\mathcal{F}}_p$ is defined by Equation $\left(6\right)$ and is not given in an explicit form. To determining whether a set of FIFs is linearly independent may not be a trivial task. Here we investigate this problem. Let $m$ be a positive integer. Suppose that, for $j=1, \cdots , m$, $f_j\in {\mathcal{F}}_p$ and is corresponding to the set $\left\{s_i;q_{ji}: i=1, \cdots , N\right\}$, where Equation $\left(8\right)$ holds with $q_i$ being replaced by $q_{ji}$, $i=1, \cdots , N$. In the following, we give a condition for $f_1, \cdots , f_m$ to be linearly independent.

Proposition 2.

If $q_{1i}, \cdots , q_{mi}$ are linearly independent on $I$ for some $i=1, \cdots , N$, then $f_1, \cdots , f_m$ are linearly independent on $I$.

Proof. 

For $a_1, \cdots , a_m\in ℝ$, ${\sum }_{j=1}^m{a}_j{f}_j$ is in ${\mathcal{F}}_p$ and is corresponding to the set $\left\{s_i;{\sum }_{j=1}^m{a}_j{q}_{ji}:i=1, \cdots , N\right\}$. By Equation $\left(13\right)$ with $n=0$, we see that ${\sum }_{j=1}^m{a}_j{f}_j$ also satisfies

$$\left({\displaystyle \sum }_{j=1}^m{a}_j{f}_j\right)\left(t\right)=s_i\left({\displaystyle \sum }_{j=1}^m{a}_j{f}_j\right)\left(L_i^{-1}\left(t\right)\right)+\left({\displaystyle \sum }_{j=1}^m{a}_j{q}_{ji}\right)\left(L_i^{-1}\left(t\right)\right), t\in I_i$$

for $i=1, \cdots , N$.

If $f_1, \cdots , f_m$ are linearly dependent on $I$, then there exist $a_1, \cdots , a_m$, not all zero, such that ${\sum }_{j=1}^m{a}_j{f}_j\equiv 0$ on $I$. This implies ${\sum }_{j=1}^m{a}_j{q}_{ji}\equiv 0$ on $I$ and hence $q_{1i}, \cdots , q_{\mathrm{m}i}$ are linearly dependent on $I$ for each $i=1, \cdots , N$. Therefore, if $q_{1i}, \cdots , q_{\mathrm{m}i}$ are linearly independent on $I$ for some $i=1, \cdots , N$, then we have a set of linearly independent FIFs $f_1, \cdots , f_m$ on $I$. □

Similarly, we have

$${\left({\displaystyle \sum }_{j=1}^m{a}_j{f}_j\right)}^{\left(n\right)}\left(t\right)=\frac{1}{a_i^n}\left\{s_i{\left({\displaystyle \sum }_{j=1}^m{a}_j{f}_j\right)}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)+{\left({\displaystyle \sum }_{j=1}^m{a}_j{q}_{ji}\right)}^{\left(n\right)}\left(L_i^{-1}\left(t\right)\right)\right\}$$

for $t\in I_i$, $i=1, \cdots , N$. If $q_{1i}^{\left(n\right)}, \cdots , q_{mi}^{\left(n\right)}$ are linearly independent on $I$ for some $i=1, \cdots , N$, then $f_1^{\left(n\right)}, \cdots , f_m^{\left(n\right)}$ are linearly independent on $I$.

3.2. Fractal-Type Positive Semi-Definite Kernels and RKHSs of FIFs

Suppose that an inner product $⟨·,·⟩$ on $\mathcal{C}\left[I\right]$ is defined. Let $m$ be a positive integer. Let $\mathbb{B}=\left\{{\psi }_1, \cdots , {\psi }_m\right\}$, where each ${\psi }_j\in {\mathcal{F}}_p$ is a FIF corresponding to a set $\left\{s_i;q_{ji}: i=1, \cdots , N\right\}$, where Equation $\left(8\right)$ holds with $q_i$ being replaced by $q_{ji}, i=1, \cdots , N$. Suppose that $q_{1i}, \cdots , q_{mi}$ are linearly independent on $I$ for some $i$. Then by Proposition 2, ${\psi }_1, \cdots , {\psi }_m$ are linearly independent on $I$. Let

$${\mathcal{F}}_{\mathbb{B}}=\mathrm{span}\left\{{\psi }_1, \cdots , {\psi }_m\right\}.$$

Then ${\mathcal{F}}_{\mathbb{B}}$ is a finite-dimensional Hilbert space with a basis $\mathbb{B}$. Let $A=\left[⟨{\psi }_i, {\psi }_j⟩\right]$. By ([6] Proposition 2.23), we see that $A$ is a positive definite matrix and hence $A$ is invertible.

Define

$$k\left(𝓌, t\right)={\displaystyle \sum }_{j=1}^m{\displaystyle \sum }_{l=1}^m{\psi }_j\left(𝓌\right){\psi }_l\left(t\right)B_{j, l}, 𝓌, t\in I,$$

(14)

where the matrix $B=\left[B_{j, l}\right]$ is the inverse of $A$. The following theorem shows that ${\mathcal{F}}_{\mathbb{B}}$ is a reproducing kernel Hilbert space and $k$ is the reproducing kernel. Similar results can be found in [27,28].

Proposition 3.

The function $k$ defined by Equation $\left(14\right)$ is positive semi-definite. The space ${\mathcal{F}}_{\mathbb{B}}$ is a finite-dimensional reproducing kernel Hilbert space with the reproducing kernel $k$.

Proof. 

Since $B$ is symmetric, let $B^{1/2}$ be the matrix such that $B^{1/2}{B}^{1/2}=B$. Let $n$ be any positive integer and let $z_1, \cdots , z_n$ be any choice of distinct points in $I$. Let $\Psi =\left[{\psi }_j\left(z_r\right)\right]$. Then for any column vector $c={\left[c_1, \cdots , c_n\right]}^T$ in ${ℝ}^n$,

$${\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{r=1}^n{c}_i{c}_{r}k\left(z_i, z_r\right)=c^T{\Psi }^{T}B\Psi c={\left(B^{1/2}\Psi c\right)}^T\left(B^{1/2}\Psi c\right)\ge 0.$$

This shows that $k$ is positive semi-definite.

Let $k_t\left(·\right)=k\left(·, t\right)$ and we write $k_t$ in the form

$$k_t\left(·\right)={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t\right)B_{j, l}\right){\psi }_j\left(·\right).$$

(15)

Then for $f={\sum }_{i=1}^m{a}_i{\psi }_i\in {\mathcal{F}}_{\mathbb{B}}$ and $t\in I$,

$$⟨f, k_t⟩={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^m{a}_i\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t\right)B_{j, l}\right)⟨{\psi }_i, {\psi }_j⟩$$

$$={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{l=1}^m{a}_i{\psi }_l\left(t\right)\left({\displaystyle \sum }_{j=1}^m{A}_{i, j}{B}_{j, l}\right)={\displaystyle \sum }_{i=1}^m{a}_i{\psi }_i\left(t\right)=f\left(t\right).$$

(16)

We also have $k\left(𝓌, t\right)=k_t\left(𝓌\right)=⟨k_t, k_{𝓌}⟩$ for $t, 𝓌\in I$.

By Equation $\left(15\right)$ with $t=t_0, \cdots , t_N$, we have

$$k_{t_i}={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t_i\right)B_{j, l}\right){\psi }_j, i=0, 1, \cdots , N.$$

(17)

Define:

$$\mathcal{H}=\mathrm{span}\left\{k_{t_0}, k_{t_1}, \cdots , k_{t_N}\right\}.$$

It is easy to see that $\mathcal{H}\subseteq {\mathcal{F}}_{\mathbb{B}}$. In fact, for $f\in \mathcal{H}$, we can write $f$ in the form $f=\sum _{i=0}^N{\alpha }_i{k}_{t_i}$ and then by Equation $\left(17\right)$,

$$f={\displaystyle \sum }_{i=0}^N{\alpha }_i\left\{{\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t_i\right)B_{j, l}\right){\psi }_j\right\}={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{l=1}^m{\alpha }_i{\psi }_l\left(t_i\right)B_{j, l}\right){\psi }_j\in {\mathcal{F}}_{\mathbb{B}}.$$

The following well-known result shows the role of the subspace $\mathcal{H}$. □

Proposition 4.

For any $g\in {\mathcal{F}}_{\mathbb{B}}$, there exists a function $f\in \mathcal{H}$ such that $f\left(t_i\right)=g\left(t_i\right)$ for $i=0, 1, \cdots , N$, and $f\le g.$

Proof. 

For $g\in {\mathcal{F}}_{\mathbb{B}}$, let $P_{\mathcal{H}}\left(g\right)$ be the orthogonal projection of $g$ on $\mathcal{H}$. Then $g-P_{\mathcal{H}}\left(g\right)$ is orthogonal to the subspace $\mathcal{H}$. By Equation $\left(16\right)$ with $t=t_i$, $i=0, 1, \cdots , N$, we have

$$g\left(t_i\right)-P_{\mathcal{H}}\left(g\right)\left(t_i\right)=⟨g-P_{\mathcal{H}}\left(g\right), k_{t_i}⟩=0.$$

This proposition is proved by choosing $f=P_{\mathcal{H}}\left(g\right)$. □

For $g\in {\mathcal{F}}_{\mathbb{B}}$, ${\Vert g\Vert }^2=⟨g,g⟩$. If $f$ is a function in $\mathcal{H}$, then ${\Vert f\Vert }^2$ can be represented by $k$. We write $f\in \mathcal{H}$ in the form $f={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}$, then

$${\Vert f\Vert }^2=⟨{\displaystyle \sum }_{i=0}^N{\mathsf{\alpha }}_i{k}_{t_i}, {\displaystyle \sum }_{j=0}^N{\mathsf{\alpha }}_j{k}_{t_j}⟩={\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{j=0}^N{\mathsf{\alpha }}_i{\mathsf{\alpha }}_{j}k\left(t_j, t_i\right).$$

(18)

3.3. RKHSs Defined by the Derivatives of Functions in $\mathbb{B}$

In Section 3.2, we establish a positive semi-definite kernel $k$ by a set of linearly independent functions ${\psi }_1, \cdots , {\psi }_m$ in ${\mathcal{F}}_p$, and show that the span of these functions is a reproducing kernel Hilbert space and $k$ is the reproducing kernel. Since all ${\psi }_i$ are functions in ${\mathcal{C}}^p\left[I\right]$ and derivatives of ${\psi }_i$ are still FIFs, it is quite nature to investigate those RKHSs which are spanned by the derivatives of ${\psi }_1, \cdots , {\psi }_m$.

Suppose $p\ge 1$ and $1\le n\le p$. If $q_{1i}^{\left(n\right)}, \cdots , q_{mi}^{\left(n\right)}$ are linearly independent on $I$ for some $i$, then ${\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}$ are linearly independent on $I$. Let ${\mathbb{B}}^{\left(n\right)}=\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}$ and

$${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}=\mathrm{span}\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}.$$

Each ${\psi }_j^{\left(n\right)}$ is a FIF corresponding to the set $\left\{\frac{s_i}{a_i^n};\frac{q_{ji}^{\left(n\right)}}{a_i^n}: i=1, \cdots , N\right\}$. Define

$${\mathcal{D}}^{n}k\left(𝓌, t\right)={\displaystyle \sum }_{j=1}^m{\displaystyle \sum }_{l=1}^m{\psi }_j^{\left(n\right)}\left(𝓌\right){\psi }_l^{\left(n\right)}\left(t\right)B_{j,l}^{\left(n\right)}, t^{\prime }, t\in I,$$

(19)

where the matrix $B^{\left(n\right)}=\left[B_{j, l}^{\left(n\right)}\right]$ is the inverse of the matrix $A^{\left(n\right)}=\left[⟨{\psi }_i^{\left(n\right)}, {\psi }_j^{\left(n\right)}⟩\right]$. We see that ${\mathcal{D}}^{n}k\left(𝓌, t\right)=\frac{{\partial }^{2n}}{\partial {𝓌}^n\partial t^n}k\left(𝓌, t\right)$. By a similar approach given in the proof of Theorem 3, we see that ${\mathcal{D}}^{n}k$ is positive semi-definite and we can write ${\left({\mathcal{D}}^{n}k\right)}_t$ in the form

$${\left({\mathcal{D}}^{n}k\right)}_t\left(·\right)={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l^{\left(\mathrm{n}\right)}\left(t\right)B_{j,l}^{\left(n\right)}\right){\psi }_j^{\left(\mathrm{n}\right)}\left(·\right).$$

(20)

For $g={\sum }_{i=1}^m{\alpha }_i{\psi }_i^{\left(n\right)}\in {\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$ and for $t\in I$,

$$⟨g, {\left({\mathcal{D}}^{n}k\right)}_t⟩={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{j=1}^m{\alpha }_i\left({\displaystyle \sum }_{l=1}^m{\psi }_l^{\left(n\right)}\left(t\right)B_{j,l}^{\left(n\right)}\right)⟨{\psi }_i^{\left(n\right)}, {\psi }_j^{\left(n\right)}⟩$$

$$={\displaystyle \sum }_{i=1}^m{\displaystyle \sum }_{l=1}^m{\alpha }_i{\psi }_l^{\left(n\right)}\left(t\right)\left({\displaystyle \sum }_{j=1}^m{A}_{i,j}^{\left(n\right)}{B}_{j, l}^{\left(n\right)}\right)={\displaystyle \sum }_{i=1}^m{\alpha }_i{\psi }_i^{\left(n\right)}\left(t\right)=g\left(t\right).$$

(21)

Therefore, we have the following theorem.

Proposition 5.

The space ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$ is a finite-dimensional reproducing kernel Hilbert space with the reproducing kernel ${\mathcal{D}}^{n}k$ defined by Equation $\left(19\right)$.

By Equation $\left(20\right)$ with $t=t_0, \cdots , t_N$, we have

$${\left({\mathcal{D}}^{n}k\right)}_{t_i}\left(·\right)={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l^{\left(n\right)}\left(t_i\right)B_{j,l}^{\left(n\right)}\right){\psi }_j^{\left(n\right)}\left(·\right), i=1, \cdots , N.$$

(22)

Define

$${\mathcal{H}}^{\left(n\right)}=\mathrm{span}\left\{{\left({\mathcal{D}}^{n}k\right)}_{t_0}, {\left({\mathcal{D}}^{n}k\right)}_{t_1},\dots ,{\left({\mathcal{D}}^{n}k\right)}_{t_N}\right\}.$$

It is easy to see that ${\mathcal{H}}^{\left(n\right)}\subseteq {\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$. In fact, for $h\in {\mathcal{H}}^{\left(n\right)}$, we can write $h={\sum }_{i=0}^N{\alpha }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}$ and then

$$h={\displaystyle \sum }_{i=0}^N{\alpha }_i\left\{{\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l^{\left(n\right)}\left(t_i\right)B_{j,l}^{\left(n\right)}\right){\psi }_j^{\left(n\right)}\right\}={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{l=1}^m{\alpha }_i{\psi }_l^{\left(n\right)}\left(t_i\right)B_{j,l}^{\left(n\right)}\right){\psi }_j^{\left(n\right)}\in {\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}.$$

The following result shows the role of the subspace ${\mathcal{H}}^{\left(n\right)}$.

Proposition 6.

For any $h\in {\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$, there exists a function $u\in {\mathcal{H}}^{\left(n\right)}$ such that $u\left(t_i\right)=h\left(t_i\right)$ for $i=0, 1, \cdots , N$, and $u\le h$.

Proof. 

For $h\in {\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$, let ${\mathit{P}}_{{\mathcal{H}}^{\left(n\right)}}$ be the orthogonal projection of $h$ on ${\mathcal{H}}^{\left(n\right)}$. Then $h-{\mathit{P}}_{{\mathcal{H}}^{\left(n\right)}}\left(h\right)$ is orthogonal to the subspace ${\mathcal{H}}^{\left(n\right)}$. By Equation $\left(21\right)$ with $t=t_i$, $i=0, 1, \cdots , N$, we have

$$h\left(t_i\right)-{\mathit{P}}_{{\mathcal{H}}^{\left(n\right)}}\left(h\right)\left(t_i\right)=⟨h-{\mathit{P}}_{{\mathcal{H}}^{\left(n\right)}}\left(h\right),{\left({\mathcal{D}}^{n}k\right)}_{t_i}⟩=0.$$

This proposition is proved by choosing $u={\mathit{P}}_{{\mathcal{H}}^{\left(n\right)}}\left(h\right)$. □

If $u$ is a function in ${\mathcal{H}}^{\left(n\right)}$, then ${\Vert u\Vert }^2$ can be represented by ${\mathcal{D}}^{n}k$. For $u\in {\mathcal{H}}^{\left(n\right)}$, we write $u$ as the form $u={\sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}$, and then

$${\Vert u\Vert }^2=⟨{\displaystyle \sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}, {\displaystyle \sum }_{j=0}^N{\beta }_j{\left({\mathcal{D}}^{n}k\right)}_{t_j}⟩={\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{j=0}^N{\beta }_i{\beta }_j{\mathcal{D}}^{n}k\left(t_j,t_i\right).$$

(23)

A function $f$ in $\mathcal{H}=\mathrm{span}\left\{k_{t_0}, k_{t_1}, \cdots , k_{t_N}\right\}$ can be written as the form $f={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}$. We have $f^{\left(n\right)}={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}$, where $k_{t_i}^{\left(n\right)}$ is the $n$th derivative of $k_{t_i}$ defined by Equation $\left(17\right)$ and it can be written as

$$k_{t_i}^{\left(n\right)}={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t_i\right)B_{j, l}\right){\psi }_j^{\left(n\right)}, i=0, 1, \cdots , N.$$

(24)

Since $f^{\left(n\right)}$ is a linear combination of $k_{t_0}^{\left(n\right)},k_{t_1}^{\left(n\right)},\dots ,k_{t_N}^{\left(n\right)}$, it is natural to consider the space

$${ℍ}^{\left(n\right)}=\mathrm{span}\left\{k_{t_0}^{\left(n\right)}, k_{t_1}^{\left(n\right)}, \cdots , k_{t_N}^{\left(n\right)}\right\}$$

when we discuss the $n$th derivative $f^{\left(n\right)}$ of $f\in \mathcal{H}$. Recall that ${\mathcal{H}}^{\left(n\right)}=\mathrm{span}\left\{{\left({\mathcal{D}}^{n}k\right)}_{t_0}, {\left({\mathcal{D}}^{n}k\right)}_{t_1},\dots ,{\left({\mathcal{D}}^{n}k\right)}_{t_N}\right\}$. Is the subspace ${\mathcal{H}}^{\left(n\right)}$ identical to the subspace ${ℍ}^{\left(n\right)}$? We discuss this question below. Here both sets of functions $\left\{{\psi }_1, \cdots , {\psi }_m\right\}$and $\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}$ are supposed to be linearly independent.

We first consider functions in ${ℍ}^{\left(n\right)}\cap {\mathcal{H}}^{\left(n\right)}$. Suppose $h\in {ℍ}^{\left(n\right)}\cap {\mathcal{H}}^{\left(n\right)}$. We can write

$$h={\displaystyle \sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}={\displaystyle \sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}.$$

(25)

Then by Equations $\left(24\right)$ and $\left(22\right)$,

$$h={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{l=1}^m{\alpha }_i{\psi }_l\left(t_i\right)B_{j, l}\right){\psi }_j^{\left(n\right)}={\displaystyle \sum }_{j=1}^m\left({\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{l=1}^m{\beta }_i{\psi }_l^{\left(n\right)}\left(t_i\right)B_{j,l}^{\left(n\right)}\right){\psi }_j^{\left(n\right)}.$$

Since ${\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}$ are linearly independent on $I$, we have

$${\displaystyle \sum }_{i=0}^N\left({\displaystyle \sum }_{l=1}^m{\psi }_l\left(t_i\right)B_{j, l}\right){\alpha }_i={\displaystyle \sum }_{i=0}^N\left({\displaystyle \sum }_{l=1}^m{\psi }_l^{\left(n\right)}\left(t_i\right)B_{j,l}^{\left(n\right)}\right){\beta }_i, j=1, \cdots , m.$$

(26)

The system Equation $\left(26\right)$ can be written in the matrix form

$$B\Psi \left(t\right)\mathit{\alpha }=B^{\left(n\right)}{\Psi }^{\left(n\right)}\left(t\right)\mathit{\beta },$$

(27)

where $\mathit{\alpha }={\left[{\alpha }_0, \cdots , {\alpha }_N\right]}^T$, $\mathit{\beta }={\left[{\beta }_0, \cdots , {\beta }_N\right]}^T$, and $\Psi \left(t\right)$, ${\Psi }^{\left(n\right)}\left(t\right)$ are $m\times \left(N+1\right)$ matrices $\left[{\psi }_l\left(t_i\right)\right]$, $\left[{\psi }_l^{\left(n\right)}\left(t_i\right)\right]$, respectively. Here $B=\left[B_{j, l}\right]$ is given in Equation $\left(14\right)$ and $B^{\left(n\right)}=\left[B_{j, l}^{\left(n\right)}\right]$ is given in Equation $\left(19\right)$. Hence $h\in {ℍ}^{\left(n\right)}\cap {\mathcal{H}}^{\left(n\right)}$ if and only if there exist vectors $\mathit{\alpha }$ and $\mathit{\beta }$ that satisfy Equation $\left(27\right)$, and in this case, $h$ is given by Equation $\left(25\right)$.

Since $B$ is invertible, we have

$$\Psi \left(t\right)\mathit{\alpha }=B^{-1}{B}^{\left(n\right)}{\Psi }^{\left(n\right)}\left(t\right)\mathit{\beta }.$$

(28)

If the rank of $\Psi \left(t\right)$ is equal to $m$, then this equation is consistent for any vector $\mathit{\beta }$. Conversely, since $B^{\left(n\right)}$ is also invertible, Equation $\left(27\right)$ implies that

$${\left(B^{\left(n\right)}\right)}^{-1}B\Psi \left(t\right)\mathit{\alpha }={\Psi }^{\left(n\right)}\left(t\right)\mathit{\beta }.$$

(29)

If the rank of ${\Psi }^{\left(n\right)}\left(t\right)$ is equal to $m$, then this equation is consistent for any vector $\alpha$. We have the following theorem.

Proposition 7.

If the ranks of $\Psi \left(t\right)$ and ${\Psi }^{\left(n\right)}\left(t\right)$ are both equal to $m$, then ${ℍ}^{\left(\mathrm{n}\right)}={\mathcal{H}}^{\left(\mathrm{n}\right)}$.

Proof. 

For any $h\in {\mathcal{H}}^{\left(n\right)}$, we write $h$ in the form $h={\sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}$. If the rank of $\Psi \left(t\right)$ is equal to $m$, then by setting $\mathit{\beta }={\left[{\beta }_0, \cdots , {\beta }_N\right]}^T$, the Equation $\left(28\right)$ has a solution $\mathit{\alpha }={\left[{\alpha }_0, \cdots , {\alpha }_N\right]}^T$, and $h$ can be written as $h={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}\in {ℍ}^{\left(n\right)}$. This shows that ${\mathcal{H}}^{\left(n\right)}\subseteq {ℍ}^{\left(n\right)}$. Conversely, for any $h\in {ℍ}^{\left(n\right)}$, we write $h$ in the form $h={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}$. If the rank of ${\Psi }^{\left(n\right)}\left(t\right)$ is equal to $m$, then by setting $\mathit{\alpha }={\left[{\alpha }_0, \cdots , {\alpha }_N\right]}^T$, the Equation $\left(29\right)$ has a solution $\mathit{\beta }={\left[{\beta }_0, \cdots , {\beta }_N\right]}^T$, and $h$ can be written as $h={\sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}\in {\mathcal{H}}^{\left(n\right)}$. This implies that ${ℍ}^{\left(n\right)}\subseteq {\mathcal{H}}^{\left(n\right)}$. □

For $f\in \mathcal{H}$, we write $f={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}$ and $f^{\left(n\right)}={\sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}\in {ℍ}^{\left(n\right)}$. If $\mathit{\beta }$ is a solution of Equation $\left(27\right)$, then we can write $f^{\left(n\right)}={\sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}\in {\mathcal{H}}^{\left(n\right)}$ and

$${\Vert f^{\left(n\right)}\Vert }^2={\displaystyle \sum }_{i=0}^N{\displaystyle \sum }_{j=0}^N{\beta }_i{\beta }_j{\mathcal{D}}^{n}k\left(t_j, t_i\right).$$

We can investigate the relationship between $k_{t_j}^{\left(n\right)}$ and ${\left({\mathcal{D}}^{n}k\right)}_{t_i}$ by Equation $\left(27\right)$. If $\mathit{\alpha }$ is the vector such that ${\alpha }_{\delta }=1$ and ${\alpha }_i=0$ for $i\ne \delta$, where $0\le \delta \le N$, then the left-hand side of Equation $\left(27\right)$ is reduced to the $\delta$-th column of the matrix $B\Psi \left(t\right)$, and if $\mathit{\beta }$ is a solution of Equation $\left(27\right)$, then

$$k_{t_{\delta }}^{\left(n\right)}={\displaystyle \sum }_{i=0}^N{\beta }_i{\left({\mathcal{D}}^{n}k\right)}_{t_i}.$$

If $\mathit{\beta }$ is the vector such that ${\beta }_{\delta }=1$ and ${\beta }_i=0$ for $i\ne \delta$, where $0\le \delta \le N$, then the right-hand side of Equation $\left(27\right)$ is reduced to the $\delta$-th column of the matrix $B^{\left(n\right)}{\Psi }^{\left(n\right)}\left(t\right)$, and if $\mathit{\alpha }$ is a solution of Equation $\left(27\right)$, then

$${\left({\mathcal{D}}^{n}k\right)}_{t_{\delta }}={\displaystyle \sum }_{i=0}^N{\alpha }_i{k}_{t_i}^{\left(n\right)}.$$

Consider the particular case $m=N+1$. If the ranks of $\Psi \left(t\right)$ and ${\Psi }^{\left(n\right)}\left(t\right)$ are both equal to $N+1$, then $\Psi \left(t\right)$ and ${\Psi }^{\left(n\right)}\left(t\right)$ are both invertible. By Equation $\left(27\right)$,

$$\begin{gathered} \mathit{\alpha }=\Psi {\left(t\right)}^{-1}{B}^{-1}{B}^{\left(n\right)}{\Psi }^{\left(n\right)}\left(t\right)\mathit{\beta } \mathrm{for} \mathrm{any} \mathrm{vector} \mathit{\beta }, \\ \mathit{\beta }={\left({\Psi }^{\left(n\right)}\left(t\right)\right)}^{-1}{\left(B^{\left(n\right)}\right)}^{-1}B\Psi \left(t\right)\mathit{\alpha } \mathrm{for} \mathrm{any} \mathrm{vector} \mathit{\alpha }. \end{gathered}$$

4. Conclusions

The theory of RKHSs and the concept of FIFs play important roles in mathematics and have variety of applications in many fields. The work given in [27,28] bridged the gap between RKHSs and FIFs. In this paper we study RKHSs of smooth FIFs $f$ and their $n$th derivatives $f^{\left(n\right)}$.

First, a linear space ${\mathcal{F}}_p$ of ${\mathcal{C}}^p$-FIF is established. Then we consider a linearly independent set of functions $\mathbb{B}=\left\{{\psi }_1, \cdots , {\psi }_m\right\}$ in ${\mathcal{F}}_p$. A condition for the independent property of functions in $\mathbb{B}$ is established. We show that the space ${\mathcal{F}}_{\mathbb{B}}=\mathrm{span}\left\{{\psi }_1, \cdots , {\psi }_m\right\}$ is a finite-dimensional RKHS with the reproducing kernel $k$ defined by Equation $\left(14\right)$. Define $\mathcal{H}=\mathrm{span}\left\{k_{t_0}, k_{t_1}, \cdots , k_{t_N}\right\}$, which is a subspace of ${\mathcal{F}}_{\mathbb{B}}$ and plays an important role in curve fitting applications. If $f$ is a function in $\mathcal{H}$, then ${\Vert f\Vert }^2$ can be represented by $k$.

To investigate the RKHS of $n$th derivatives of functions in ${\mathcal{F}}_{\mathbb{B}}$, we define ${\mathbb{B}}^{\left(n\right)}=\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}$ and ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}=\mathrm{span}\left\{{\psi }_1^{\left(n\right)}, \cdots , {\psi }_m^{\left(n\right)}\right\}$. ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$ is a finite-dimensional RKHS with the reproducing kernel ${\mathcal{D}}^{n}k$ defined by Equation $\left(19\right)$. For $f\in \mathcal{H}$, $f^{\left(n\right)}\in {ℍ}^{\left(n\right)}=\mathrm{span}\left\{k_{t_0}^{\left(n\right)}, k_{t_1}^{\left(n\right)}, \cdots , k_{t_N}^{\left(n\right)}\right\}$. We also define a subspace ${\mathcal{H}}^{\left(n\right)}$ of ${\mathcal{F}}_{{\mathbb{B}}^{\left(n\right)}}$, where ${\mathcal{H}}^{\left(n\right)}=\mathrm{span}\left\{{\left({\mathcal{D}}^{n}k\right)}_{t_0}, {\left({\mathcal{D}}^{n}k\right)}_{t_1},\dots ,{\left({\mathcal{D}}^{n}k\right)}_{t_N}\right\}$. We prove that if the ranks of $\Psi \left(t\right)$ and ${\Psi }^{\left(n\right)}\left(t\right)$ are both equal to $m$, then ${ℍ}^{\left(n\right)}={\mathcal{H}}^{\left(n\right)}$. Here $\Psi \left(t\right)=\left[{\psi }_l\left(t_i\right)\right]$ and ${\Psi }^{\left(n\right)}\left(t\right)=\left[{\psi }_l^{\left(n\right)}\left(t_i\right)\right]$ are both $m\times \left(N+1\right)$ matrices. If $f$ is a function in $\mathcal{H}$, then ${\Vert f^{\left(n\right)}\Vert }^2$ can be represented by ${\mathcal{D}}^{n}k$. Note that $f^{\left(n\right)}$ can be represented by ${\left({\mathcal{D}}^{n}k\right)}_{t_0}, {\left({\mathcal{D}}^{n}k\right)}_{t_1},\dots ,{\left({\mathcal{D}}^{n}k\right)}_{t_N}$ and also by $k_{t_0}^{\left(n\right)}, k_{t_1}^{\left(n\right)}, \cdots , k_{t_N}^{\left(n\right)}$, the two representations are connected by the Formula $\left(27\right)$. Moreover, it is clear to write each ${\left({\mathcal{D}}^{n}k\right)}_{t_i}$ as a linear combination of $k_{t_0}^{\left(n\right)}, k_{t_1}^{\left(n\right)}, \cdots , k_{t_N}^{\left(n\right)}$, and to write each $k_{t_i}^{\left(n\right)}$ as a linear combination of ${\left({\mathcal{D}}^{n}k\right)}_{t_0}, {\left({\mathcal{D}}^{n}k\right)}_{t_1},\dots ,{\left({\mathcal{D}}^{n}k\right)}_{t_N}$.

In this paper we establish a finite-dimensional RKHS spanned by a set of linearly independent FIFs, and the RKHS spanned by the $n$th derivatives of these basis functions. An important subspace $\mathcal{H}$ is introduced, and the subspace of the $n$th derivative of $f$ in $\mathcal{H}$ is investigated. We believe that results established in this paper enrich the theory of RKHSs and FIFs, and may have applications in many fields.