We exhibit the necessary range for which functions in Sobolev spaces L p s $L^s_p$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such Battle–Lemarié wavelets. also consider natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work Seeger Ullrich, where analogous results were established Haar system.

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new...

A method for the design of Fast Haar wavelet for signal processing & image processing has been proposed. In the proposed work, the analysis bank and synthesis bank of Haar wavelet is modified by using polyphase structure. Finally, the Fast Haar wavelet was designed and it satisfies alias free and perfect reconstruction condition. Computational time and computational complexity is reduced in Fas...

A wavelet method to the solution for time-fractional partial differential equation, by which combining with Haar wavelet and operational matrix to discretize the given functions efficaciously. The time-fractional partial differential equation is transformed into matrix equation. Then they can be solved in the computer oriented methods. The numerical example shows that the method is effective.

Can we characterize the wavelets through linear transformation? the answer for this question is certainly YES. In this paper we have characterized the Haar wavelet matrix by their linear transformation and proved some theorems on properties of Haar wavelet matrix such as Trace, eigenvalue and eigenvector and diagonalization of a matrix.

We address $L^p(\mu)\to L^p(\lambda)$ bounds for paraproducts in the Bloom setting. introduce certain “sparse BMO” functions associated with sparse collections no infinitely increasing chains, and use these to express operators as sums of martingale transforms – essentially, Haar multipliers well obtain an equivalence norms between $\mathcal{A}\_{\mathcal{S}}$ compositions $\Pi^\*\_a\Pi\_b$.

The representation of a general Calder\'on--Zygmund operator in terms dyadic Haar shift operators first appeared as tool to prove the $A_2$ theorem, and it has found number other applications. In this paper we new theorem by using smooth compactly supported wavelets place functions. A key advantage is that achieve faster decay expansion when kernel additional smoothness.