Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials
نویسندگان
چکیده
منابع مشابه
Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials.
Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. These functions are generated from gradients of Zernike polynomials, made orthonorm...
متن کاملOrthonormal vector polynomials in a unit circle, Part II : Completing the basis set.
Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. Previously, we have developed a basis of functions generated from gradients of Zern...
متن کاملStrehl ratio and amplitude-weighted generalized orthonormal Zernike-based polynomials.
The concept of orthonormal polynomials is revisited by developing a Zernike-based orthonormal set for a non-circular pupil that is transmitting an aberrated, non-uniform field. We refer to this pupil as a general pupil. The process is achieved by using the matrix form of the Gram-Schmidt procedure on Zernike circle polynomials and is interpreted as a process of balancing each Zernike circle pol...
متن کاملAsymptotic Behavior of Polynomials Orthonormal on a Homogeneous Set
Abstract. Let E be a homogeneous compact set, for instance a Cantor set of positive length. Further let σ be a positive measure with supp(σ) = E. Under the condition that the absolutely continuous part of σ satisfies a Szegö–type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to σ. For the special case that E consists of a fi...
متن کاملStieltjes-type Polynomials on the Unit Circle
Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Optics Express
سال: 2007
ISSN: 1094-4087
DOI: 10.1364/oe.15.018014