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The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of

Skew circulant and circulant matrices have became important tools in solving various differential equations. Bertaccini and Ng [

Skew circulant matrices have important applications in various disciplines including image processing, communications, signal processing, encoding, solving Toeplitz matrix problems, preconditioner, and solving least squares problems. They have been put on firm basis with the work of Davis [

Besides, there are several papers on the circulant operator and circulant algebra. Wilde [

In passing, skew-circulant operator and algebra were only used in [

In Section

In Section

In Section

In Section

An

Let

Then, through the eigenvalues of the matrix

If

By calculation, these matrices

We have seen that every skew-circulant matrix,

If the function

By composition, we gain that

The function

Let

Since

Similarly, we can obtain

And, by calculation, we have

Finally, we gain

Synthesizing Propositions

The algebras generated by

If

According to [

We thus have

Then, we complete the proof of this conclusion.

For any linear entire function

For each

Equation (

If the function

All these equations (

Linear combinations of the operators

If the operators

Through simple calculation, we can get properties (

By all accounts, a function

In addition,

By (

For

Hence, from (

Let

From what has been discussed above, we gain the following theorem.

The

The projections

We can prove (

This result can be rewritten in the form

Finally, it can be shown that

For each

By (

Suppose there exists another function

Now the results of all this are as follows: let

Suppose that

Let

From (

From (

Consider the set

If

Note that, if

In the same way, (

We prove that the set of skew-circulants with complex entries has an idempotent basis. This paper displays algebras of operators which are isomorphic to the algebra of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research is supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11301251) and the AMEP of Linyi University, China.