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Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

A multiscaling equation in the Fourier domain accommodates a trigonometric matrix polynomial. This trigonometric matrix polynomial is known as the symbol function. The existence and properties of a multiscaling function, which is the solution of a multiscaling equation, depend on the symbol function. It is possible to construct symbol functions corresponding to compactly supported and symmetric multiscaling functions from standard pairs. A standard pair carries the spectral information about the symbol function. In this paper, we briefly explain the construction of compactly supported and symmetric multiscaling functions and the corresponding mulitwavelets using standard pairs. We derive the necessary as well as sufficient condition, on the eigenspace of the square matrix in the standard pair, for the existence of a symbol function corresponding to a multiscaling equation with a compactly supported solution. We create a pseudo bi-orthogonal pair of symmetric and compactly supported multiscaling functions and the corresponding multiwavelets using standard pairs.

A multiwavelet is typically constructed starting from a vector-valued function satisfying a matrix refinement equation. The approximation order of such a refinable function vector is related to the sum rules of order $p$ satisfied by the corresponding refinement mask. A refinable function vector can be obtained using cascade algorithm by constructing a refinement mask which satisfies the sum rules of order $1.$ A standard pair associated with a refinement mask gives information about its spectral properties. In this paper, we present a procedure for constructing refinement masks satisfying the sum rules of order $1$, starting from standard pairs. How this helps in the construction of asymmetric multiwavelets using standard pairs is illustrated through examples. A sufficient condition on a standard pair and a necessary and sufficient condition on a left standard pair are established so that the corresponding refinement mask satisfies the sum rules of order $1$.

In this paper, using standard pairs, we present a method to construct symbols satisfying the sum rules of order $p$ for any given $p$. It is shown that using matrix polynomial theory symbols, symmetric or non-symmetric, satisfying the sum rules of order $p$ can be constructed efficiently. The construction is illustrated using various examples.