We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods.
Cubic spline, as the most commonly used spline in practice, is a fundamental approximation tool [
Mathematically, cubic spline interpolation is often introduced as follows. Let
However, we often meet two troubles in the practical applications of cubic spline interpolation. The first trouble is that we cannot obtain the precise function values in (
To deal with the troubles, in this paper, we give two new effective cubic spline based methods for reconstructing
We organize the remainder of this paper as follows. In Section
We assume that the nodes in (
The values of



Else  





0 


0 

0 




0 
Using twopoint numerical differentiation formula, we have
The approximate boundary derivatives and their errors (I).
Twopoint results  Threepoint results  



















The approximate boundary derivatives and their errors (II).
Fivepoint results  















We study the following noisy lacunary cubic spline interpolation (
Let
By using the given function data, we can directly get a cubic spline
The method is very simple and effective method for noisy data because it avoids using approximate boundary derivatives and also avoids solving the linear system (
We denote (
Add column one to column three and also add column
Let
The bounds of
Twopoint results  Threepoint results  Fivepoint results  









Consider
Consider
Consider
Because of property (
By the nonnegativity and partition of unity of cubic Bsplines, for
For
For
Let
First of all, for
Let
These results follow from the traditional cubic spline interpolation error theory [
From (
Let
By (
Let
We first prove (
Next, we prove (
Finally, we prove (
In this section, we perform numerical tests by Matlab. The following examples
In every numerical test, the mesh size
In Tables
Numerical results of
Method  I1  I2  I3  II  CSM [  





















 






















 






















 























Numerical results of
Method  I1  I2  I3  II  CSM [  





































































 























Generally, the maximum absolute errors
When
It is very reasonable to compare our methods with the cubic spline method (CSM) in [
When
The explicit error bounds for a noisy lacunary cubic spline interpolation and a simple noisy cubic spline quasi interpolation are well studied in this paper; see Theorems
The main contributions of the paper include (i) studying two new methods to approximate a function and its first order and second order derivatives from the given noisy data and (ii) analyzing the explicit error bounds for the methods.
The main advantages of our new methods include the following: (i) they are very simple; (ii) they are not only applicable to noisy data but also applicable to exact data; (iii) Method I2 and Method I3 have better performance in function approximation and first order derivative approximation than other methods; Method I3 and Method II have better performance in second order derivative approximation than other methods.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors appreciate the reviewers and editors for their careful reading, valuable suggestions, and timely review and reply.