An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition...

1 Review 1.1 Setup In the previous lecture, we considered an apparatus consisting of three planes: • an aperture plane. Coordinates on the aperture plane are denoted by x and y. • a transform plane. Coordinates on the transform plane are denoted by X and Y. • an image plane. Coordinates on the image plane are denoted by x and y. We defined the spatial frequencies, µ = k f X ν = k f Y where f is...

Let G be a locally compact abelian group with dual group Ĝ. The Hausdorff–Young theorem states that if f ∈ Lp(G), where 1 ≤ p ≤ 2, then its Fourier transform Fp(f) belongs to Lq(Ĝ) (where 1 p + 1 q = 1) and ||Fp(f)||q ≤ ||f ||p. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G b...

This paper shows that binary decision diagrams (BDDs) and their generalizations are not only representations of switching and integer-valued functions, but also Fourier-like series expansions of them. Furthermore, it shows that edge-valued binary decision diagrams (EVBDDs) are related to arithmetic transform decision diagrams (ACDDs), which are the integer counterparts of the functional decisio...

Let f : M → G(m, n) be a harmonic map from surface into complex Grassmann manifold. In this paper, some sufficient conditions for the harmonic sequence generated by f to have degenerate ∂-transform or ∂-transform are given.

Both the Walsh system and the trigonometric system are systems of characters on a compact abelian group. This explains that many of the results in the theory of those systems are parallel. However, those similarities do usually not extend to the case when the systems are compared directly. So it is known that the Walsh system in the Walsh-Paley order and the trigonometric system are not equival...

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients {an} and the continued-fraction coefficients {bn}. In many instances it turns out that this nonlinear relation transforms a complicated sequence {an} into a very simple one {bn}. We illustrate this simplification in the context of graph combinatorics. PACS numbers: 02.90...

This paper presents an algebraic approach to construct M-band orthonormal wavelet bases with perfect reconstruction. We first derive the system of constraint equations of M-band filter banks, and then an algebraic solution based on matrix decomposition is developed. The structure of the solutions is presented, and practical construction procedures are given. By using this algebraic approach, so...

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform f̂ of continuous or generalized functions f . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of f̂ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sens...