Reproducing Kernel Functions

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.


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.

Reproducing kernel spaces
In this section, we define some useful reproducing kernel functions .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 K Á; t ðÞ : 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 is a Hilbert space.The inner product and the norm in W 3 2 0; 1 ½ are defined by 0 ðÞ ɡ i ðÞ 0 ðÞþ respectively.Thus, the space W 3 2 0; 1 ½ is a reproducing kernel space, that is, for each fixed y ∈ 0; 1 ½ and any v ∈ W 3 2 0; 1 ½ , there exists a function R y such that vy ðÞ¼ vx ðÞ ; R y x ðÞ and similarly, we define the space The inner product and the norm in T 3 2 0; 1 ½ are defined by respectively.The space T 3 2 0; 1 ½ is a reproducing kernel Hilbert space, and its reproducing kernel function r s is given by [1] as and the space is a Hilbert space, where the inner product and the norm in G respectively.The space G 1 2 0; 1 ½ is a reproducing kernel space, and its reproducing kernel function Q y is given by [1] as Theorem 1.1.The space W 3 2 0; 1 ½ is a complete reproducing kernel space whose reproducing kernel R y is given by R y x ðÞ¼ where through iterative integrations by parts for (11), we have Then by (11), we obtain when Since we have From ( 14) and ( 19), the unknown coefficients c i y ðÞand d i y ðÞ i ¼ 1; 2; …; 6 ðÞ can be obtained.Thus, R y is given by Differential Equations -Theory and Current Research Now, we note that the space given in [1] as is a binary reproducing kernel Hilbert space.The inner product and the norm in W Ω ðÞ are defined by respectively.
Theorem 1.2.The W Ω ðÞ is a reproducing kernel space, and its reproducing kernel function is Similarly, the space is a binary reproducing kernel Hilbert space.The inner product and the norm in c W Ω ðÞ are defined by [1] as respectively.c W Ω ðÞ is a reproducing kernel space, and its reproducing kernel function G y;s ðÞ is Definition 1.3.
|ux ðÞ ,u 0 x ðÞ ,u 0 0 x ðÞ , are absolutely continuous in 0; 1 ½ The inner product and the norm in W The space W 3 2 0; 1 ½ is a reproducing kernel space, that is, for each fixed y ∈ 0; 1 ½ and any ux ðÞ ∈ W 3 2 0; 1 ½ , there exists a function R y x ðÞsuch that uy ðÞ¼ ux ðÞ ; R y x ðÞ The inner product and the norm in W and The space W 1 2 0; 1 ½ is a reproducing kernel space, and its reproducing kernel function T x y ðÞis given by Theorem 1.5.The space W 3 2 0; 1 ½ is a complete reproducing kernel space, and its reproducing kernel function R y x ðÞcan be denoted by R y x ðÞ¼ where then by (31), we have the following equation: and From ( 33)-( 36), the unknown coefficients c i y ðÞand d i y ðÞ i ¼ 1; 2; …; 6 ðÞ can be obtained.Thus R y x ðÞis given by R y x ðÞ¼ x 3 y 2 À 5 936 x 5 y 4 À 1 18720 x 5 y 5 À 1 624 x 5 y 2 À 1 1872 x 5 y 3 ,x ≤ y The inner product and the norm in W The space W 4 2 0; 1 ½ is a reproducing kernel space, that is, for each fixed.
y ∈ 0; 1 ½ and any vx ðÞ ∈ W 4 2 0; 1 ½ , there exists a function R y x ðÞsuch that vy ðÞ¼ vx ðÞ ; R y x ðÞ Similarly, we define the space The inner product and the norm in W 2 2 0; T ½ are defined, respectively, by Thus, the space W 2 2 0; T ½ is also a reproducing kernel space, and its reproducing kernel function r s t ðÞcan be given by and the space (43) The space W 2 2 0; 1 ½ is a reproducing kernel space, and its reproducing kernel function Q y x ðÞis given by Similarly, the space W 1 2 0; T ½ is defined by The inner product and the norm in W 1 2 0; T ½ are defined, respectively, by The space W 1 2 0; T ½ is a reproducing kernel space, and its reproducing kernel function q s t ðÞis given by Further, we define the space W Ω ðÞ as and the inner product and the norm in W Ω ðÞ are defined, respectively, by Now, we have the following theorem: Theorem 1.6.The space W 4 2 0; 1 ½ is a complete reproducing kernel space, and its reproducing kernel function R y x ðÞcan be denoted by R y x ðÞ¼ where then by (54), we obtain the following equation: (59)

Since
Reproducing Kernel Functions http://dx.doi.org/10.5772/intechopen.75206 Definition 2.1 (reproducing kernel).Let E be a nonempty set.A function K : E Â E !ℂ is called a reproducing kernel of the Hilbert space H if and only if ,