arXiv subjects
Publications and source records attributed to Noufal Asharaf.
Coorbit spaces provide a rigorous framework for the assessment of the approximation theoretic properties of generalized wavelet systems. It is therefore useful to understand when two different wavelet systems give rise to the same scales of coorbit spaces. This paper provides an exhaustive answer to this question for the case of continuous wavelet transforms associated with matrix groups in dimension two.
This manuscript focuses on the construction of compactly supported dual Gabor frames in $L^2(\mathbb{R})$. The performance of the constructed dual frames is analysed for Gabor systems generated by B-splines and exponential B-splines of orders 2 and 3. The reconstruction performance of these dual windows is evaluated using the average mean square error (AMSE) for standard one-dimensional benchmark signals. For two-dimensional data, image reconstruction experiments are carried out using tensor-product Gabor frames, and the reconstruction accuracy is also assessed using AMSE. Using the duality condition for Gabor systems \cite{jan}, several alternate dual windows with finite support are constructed under suitable assumptions, such as the partition of unity property. Additional dual windows can also be obtained from an existing dual window. The canonical dual window admits an explicit expression that avoids direct inversion of the frame operator and yields reconstruction errors close to numerical precision. The constructed non-canonical compactly supported duals also exhibit stable and competitive reconstruction performance. These findings indicate that compactly supported dual windows based on B-spline and exponential B-spline generators provide effective and practical alternatives for signal and image processing applications, particularly in situations where compact support and computational efficiency are important.
Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are examined in this work. The classical NIEP restricted to non-negative matrices having a persymmetric structure is the persymmetric non-negative inverse eigenvalue problem (PNIEP). We resolve the open problem stated in \cite{Ana}, and furthermore, equivalence of the PNIEP and NIEP is established for trace-zero spectra of five complex numbers. Also we obtain new sufficient conditions for the realizability of certain classes of spectra with non-negative persymmetric matrix realization. Based on generic structural features of persymmetric matrices and their characteristic polynomials, our method is constructive in nature. The effects of perturbations on imaginary part of complex eigenvalues are also analyzed and some perturbation results are derived.
The time-fractional Black-Scholes equation (TFBSE) is intended to price the options for which the underlying price fluctuates within a correlated fractal transmission system. Although the TFBSE is an influential approach for grasping the long-term memory traits of financial markets, the non-local nature of fractional derivatives makes significant challenges in finding an accurate solution. We perform an efficient use of the differential quadrature method (DQM) based on modified cubic B-splines to solve the TFBSE governing European options. This paper constructs an algorithm by the combination of time fractional discretization using the finite difference method $L1$ and space discretization using the modified cubic B-spline-based differential quadrature method. Uniform meshes are considered for the discretization of both temporal and spatial domains. Theoretical stability has been established by finding an estimate for the maximum norm of the inverse operator regardless of the involvement of mesh parameters. We trigger the Neumann series theorem to obtain a uniform bound for the inverse operator under reasonable conditions on the mesh parameters. The numerical illustrations show that this implicit numerical method exhibits a fourth-order convergence in the space direction and the order $2-α$ in time. Moreover, we observe an enhancement in order of spatial convergence whenever $α$ tends to $0$. The results obtained are then compared with existing popular techniques to demonstrate the accuracy of modified cubic B-spline-based DQM.