# B-splines on the Heisenberg group

@inproceedings{Das2021BsplinesOT, title={B-splines on the Heisenberg group}, author={Santi Rani Das and Peter R. Massopust and Radha Ramakrishnan}, year={2021} }

In this paper, we introduce a class of B-splines on the Heisenberg group H and study their fundamental properties. Unlike the classical case, we prove that there does not exist any sequence {αn}n∈N such that L(−n.−n 2 ,−αn)φn(x, y, t) = L(−n.−2 ,−αn)φn(−x,−y,−t), for n ≥ 2, where L(x,y,t) denotes the left translation on H. We further investigate the problem of finding an equivalent condition for the system of left translates to form a frame sequence or a Riesz sequence in terms of twisted…

## References

SHOWING 1-10 OF 22 REFERENCES

### Shift-invariant Spaces with Countably Many Mutually Orthogonal Generators on the Heisenberg group

- Mathematics
- 2017

Let $E(\mathscr{A})$ denote the shift-invariant space associated with a countable family $\mathscr{A}$ of functions in $L^{2}(\mathbb{H}^{n})$ with mutually orthogonal generators, where…

### Riesz sequences of translates and generalized duals with support on [0, 1]

- Mathematics
- 2006

If the integer translates of a function ø with compact support generate a frame for a subspace W of L2(ℝ),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis…

### Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

- Mathematics
- 2014

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let j, j 2 J, be a countable family of closed,…

### Frames generated by compact group actions

- Mathematics
- 2015

Let $K$ be a compact group, and let $\rho$ be a representation of $K$ on a Hilbert space $\mathcal{H}_\rho$. We classify invariant subspaces of $\mathcal{H}_\rho$ in terms of range functions, and…

### Orthonormality of wavelet system on the Heisenberg group

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2019

### Shift-invariant system on the Heisenberg Group

- Mathematics
- 2020

In this paper, a shift-invariant system of the form
$$\{L_{a\nu }g_s:\nu \in \varGamma ,s\in {{\mathbb {Z}}}\}$$
for
$$a>0$$
is studied on the Heisenberg group
$${{\mathbb {H}}^n}$$
, where …

### Interpolation and Approximation with Splines and Fractals

- Philosophy
- 2010

This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and…

### Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

- MathematicsIndagationes Mathematicae
- 2019