We have designed a cubic spline wavelet-like decomposition for the Sobolev space H (I) where I is a bounded interval. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT will map discrete samples of a function to its wavelet expansion coefficients in at most 7N log N operations. Using this transform, we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations (PDEs). Then, we test the efficiency of the DWT and apply the collocation method to solve linear and nonlinear PDEs.

Abstract. Let • • • C K_ ] c Vq c Vx c • • • be a multiresolution analysis ofL2 generated by the mth order 5-spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces ... , W_ \, Wo, Wx, ... . Consequently, the two finite sequences that describe the two-scale relations of Nm(x) and i//m(x) in terms of Nm(2x -j), ;6Z, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases {Nm(x -j)} and {y/m(x -j)} , relative to {Nm(x -j)} and {y/m(x -j)}, respectively.

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In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ andφ in L 2 (R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ jk := 2 j/2 ψ(2 j • − k) (j, k ∈ Z) form a Riesz basis for L 2 (R). If, in addition, φ lies in the Sobolev space H m (R), then the derivatives 2 j/2 ψ (m) (2 j • − k) (j, k ∈ Z) also form a Riesz basis for L 2 (R). Consequently, {ψ jk : j, k ∈ Z} is a stable wavelet basis for the Sobolev space H m (R). The pair of φ andφ are not required to be biorthogonal or semi-orthogonal. In particular, φ andφ can be a pair of B-splines. The added flexibility on φ andφ allows us to construct wavelets with relatively small supports.

Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.