The well-known first kind Chebyshev polynomials are orthogonal in the interval with respect to the weight-function and can be simply determined with the help of the recurrence formula [1]
(2.1)
Therefore, the exponential Chebyshev (EC) functions are recently defined in a similar fashion as follows [27].
Let
be a function space with the weight function . We also assume that, for a nonnegative integer n, the n th derivative of a function is also in . Then an EC polynomial can be given by
where .
This definition leads to the three-term recurrence equation for EC polynomials
(2.2)
This definition also satisfies the orthogonality condition [27]
(2.3)
where and is the Kronecker function.
Double EC functions
Basu [20] has given the product which is a form of bivariate Chebyshev polynomials. Mason et al. [5] and Doha [11] have also mentioned a Chebyshev polynomial expression for an infinitely differentiable function defined on the square by
where and are Chebyshev polynomials of the first kind, and the double primes indicate that the first term is ; and are to be taken as and for , respectively.
Definition
Based on Basu’s study, now we introduce double EC polynomials in the following form:
(2.4)
where , are EC polynomials defined by
Recurrence relation The polynomial satisfies the recurrence relations
If the function is continuous throughout the whole infinite domain , then the ’s are biorthogonal with respect to the weight function
(2.7)
and we have
(2.8)
Multiplication is said to be of higher order than if . Then the following result holds:
Function approximation
Let be an infinitely differentiable function defined on the square . Then it may be expressed in the form
(2.10)
where
(2.11)
If in Eq. (2.10) is truncated up to the m th and n th terms, then it can be written in the matrix form
(2.12)
with is a EC polynomial matrix with entries ,
(2.13)
and A is an unknown coefficient vector,
(2.14)
Matrix relations of the derivatives of a function
th-order partial derivative of can be written as
(2.15)
and its matrix form is
(2.16)
where , and
Proposition 1 Let and th-order derivative be given by (2.12) and (2.16), respectively. Then there exists a relation between the double EC coefficient row vector and th-order partial derivatives of the vector of size as
(2.17)
where and are operational matrices for partial derivatives given in the following forms:
and
Here, I and O are identity and zero matrices, respectively, and T denotes the usual matrix transpose.
Proof Taking the partial derivatives of , and both sides of the recurrence relation (2.5) with respect to x, we get
and
(2.20)
By using the relations (2.18)-(2.20) for the elements of the matrix of partial derivatives can be obtained from the following equalities:
(2.21)
Similarly, taking the partial derivatives of , and both sides of the recurrence relation (2.6) with respect to y, respectively, we write
and
(2.24)
Then with the help of the relations (2.22)-(2.24), the elements of the matrices of partial derivatives can be obtained from
(2.25)
We have noted here that for and for .
From (2.21) and (2.25), the following equalities hold for and
(2.26)
and
(2.27)
where and and I denotes identity matrix.
Then utilizing the equalities in (2.26) and (2.27), the explicit relation between the double EC polynomial row vector and those of its derivatives has been proved as follows:
or
□
Remark .
Corollary From Eqs. (2.16) and (2.17), it is clear that the derivatives of the function are expressed in terms of double EC coefficients as follows:
(2.28)