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The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.

In this paper, we show that the univariate interpolatory subdivision schemes with the optimal approximation order, which are the D-D interpolatory ones, can be obtained from the cubic B-spline scheme. Such schemes are derived in this paper by suitably using the push-back operation, a connection between the approximating and interpolatory subdivision. Then, the process to obtain the D-D schemes is generalized to the bivariate case and bivariate interpolatory schemes with the optimal approximation order are generated based on the triangular meshes. Compared with the existing bivariate interpolatory schemes with the optimal approximation order, the newly constructed ones own some advantages, such as a better performance in terms of the number of nonzero coefficients in the mask. Several examples are given and explicitly computed.

When a Fourier series is used to approximate a function with a jump discontinuity, the Gibbs phenomenon always exists. This similar phenomenon exists for wavelets expansions. Based on the Gibbs phenomenon of a Fourier series and wavelet expansions of a function with a jump discontinuity, in this paper, we consider that a Gibbs phenomenon occurs for the p-ary subdivision schemes. Similar to the method of (Appl. Math. Lett. 76:157–163, 2018), we generalize the results about the stationary binary subdivision schemes in (Appl. Math. Lett. 76:157–163, 2018) to the case of p-ary subdivision schemes. By considering the masks of subdivision schemes, we obtain a sufficient condition to determine whether there exists a Gibbs phenomenon for p-ary subdivision schemes in the limit function close to the discontinuous point. This condition consists of the positivity of the partial sums of the values of the masks. By applying this condition, we can avoid the Gibbs phenomenon for p-ary subdivision schemes near discontinuity points. Finally, some examples in classical subdivision schemes are given to illustrate the results in this paper.

In this paper, a family of non-stationary combined ternary $5$-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order $4$, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate $C^{2}$ interpolating limit curves and $C^{4}$ approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order $5$.

In this paper, a non-stationary combined subdivision scheme is presented, which can unify several existing non-stationary approximating and interpolatory subdivision schemes. This scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case, first formalized by Maillot & Stam using a push-back operator, to the non-stationary case. For such a combined scheme, we investigate its C l convergence and exponential polynomial generation/reproduction property and get that it can reach C 4 degree of smoothness and generate/reproduce certain exponential polynomials with suitable choices of the parameters. Besides, we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials. The performance of our new schemes is illustrated by several numerical examples.

In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

In this paper, a non-stationary Catmull–Clark subdivision scheme is presented which enables the user to change the shape of the surface obtained from a given initial mesh. Such a non-stationary subdivision scheme is constructed by taking tensor product of a properly modified cubic exponential B-spline scheme in the regular mesh and giving suitable rules in the neighborhoods of extraordinary points. For this new scheme, we show that it can generate tangent plane continuous surfaces. Besides, we propose a generalization, which could locally control the limit surface for the purpose of generating more flexible surfaces and providing some features control. Some examples are given to show the performance of the new schemes.