Using normalized Hermite functions, we construct bases in the space of square integrable functions on unit circle (L2(C)) and l2(Z), which are related to each other by means Fourier transform discrete transform. These relations unitary. The construction orthonormal requires use Gramm–Schmidt method. On both spaces, have provided ladder operators with same properties as for one-dimensional quant...

Besides many advantages of wavelet transform, it has several drawbacks, e.g. ringing, shift variance, aliasing and lack of directionality. Some of them can be eliminated by using wavelet packet transform, stationary wavelet transform, complex wavelet transform, adaptive directional lifting-based wavelet transform, or adaptive wavelet filter banks that use either L2 or L1 norm. This paper contai...

Using a generalized spherical mean operator, we obtain the generalizationof Titchmarsh's theorem for the Dunkl transform for functions satisfyingthe Lipschitz condition in L2(Rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

In Hilbert space L2(Rn), we prove the equivalence between the mod-ulus of smoothness and the K-functionals constructed by the Sobolev space cor-responding to the Fourier transform. For this purpose, Using a spherical meanoperator.

A. In this article we prove that the n linear operator whose symbol is the characteristic function of the simplex ∆n = ξ1 < ... < ξn is bounded from L2 × ... × L2 into L2/n, generalizing in this way our previous work on the “bi-est” operator [12], [13] (which corresponds to the case n = 3) as well as the Lacey Thiele theorem on the bi-linear Hilbert transform [7], [8] (which corresponds ...

The quaternion wavelet transform is a new multiscale analysis tool. Firstly, this paper studies the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatial L2 R2 , proves and presents quaternion wavelet’s scale basis function and wavelet basis function concepts in spatial scale space L2 R2;H , and studies quaternion wavelet transform structure. Fina...

We introduce a method to construct large classes of MSF wavelets of the Hardy space H2(R) and symmetric MSF wavelets of L2(R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L2(R) and H2(R). We also enumerate all symmetric wavelets of L2(R) with at most three intervals in the positive axis as well as 3-interval wavelet sets o...

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic L2-estimates and the invertibility of the Fourier transform on L2(Rn) and S′(Rn).

Using recent results where we constructed refinable functions for composite dilations, we construct interpolating multiscale decompositions for such systems. Interpolating wavelets are a well-known construction similar to the usual L2-theory but with L2-projectors onto the scaling spaces replaced by L∞projectors defined by an interpolation procedure. A remarkable result is that this much simple...