Wavelets by orthogonal rational kernels
نویسنده
چکیده
Suppose f k g n k=0 is an orthonornal basis for the function space L n of polynomi-als or rational functions of degree n with prescribed poles. Suppose n = 2 s and set V s = L n. Then k n (z; w) = P n k=0 k (z) k (w), is a reproducing kernel for V s. For xed w, such reproducing kernels are known to be functions localized in the neighborhood of z = w. Moreover, by an appropriate choice of the parameters f nk g n k=0 , the functions f' n;k (z) = k n (z; nk)g n k=0 will be an orthogonal basis for V s. The orthogonal complement W s = V s+1 V s is spanned by the functions f n;k (z) = l n (z; nk)g n1 k=0 for an appropriate choice of the parameters f nk g n1 k=0 where l n = k n+1 k n is the reproducing kernel for W s. These observations form the basic ingredients for a wavelet type of analysis for orthogonal rational functions on the unit circle or the real line with respect to an arbitrary probability measure. Abstract Suppose f k g n k=0 is an orthonornal basis for the function space L n of polynomi-als or rational functions of degree n with prescribed poles. Suppose n = 2 s and set V s = L n. Then k n (z; w) = P n k=0 k (z) k (w), is a reproducing kernel for V s. For xed w, such reproducing kernels are known to be functions localized in the neighborhood of z = w. Moreover, by an appropriate choice of the parameters f nk g n k=0 , the functions f' n;k (z) = k n (z; nk)g n k=0 will be an orthogonal basis for V s. The orthogonal complement W s = V s+1 V s is spanned by the functions f n;k (z) = l n (z; nk)g n1 k=0 for an appropriate choice of the parameters f nk g n1 k=0 where l n = k n+1 k n is the reproducing kernel for W s. These observations form the basic ingredients for a wavelet type of analysis for orthogonal rational functions on the unit circle or the real line with respect to an arbitrary probability measure.
منابع مشابه
Wavelets generated by using discrete singular convolution kernels
This paper explores the connection between wavelet methods and an efficient computational algorithm—the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cas...
متن کاملOrthogonal Non{Bandlimited Wavelets on the Sphere
This paper introduces orthogonal non{bandlimited wavelets on the sphere with respect to a certain Sobolev space topology. The construction of those kernels is based on a clustering of the index set N = f(n; k) 2 N0 Zj n k ng associated to the system of spherical harmonics fYn;kg(n;k)2N . The wavelets presented here form reproducing kernels of the spans of the clustered harmonics. More explicitl...
متن کاملMultivariate periodic wavelet analysis
General multivariate periodic wavelets are an efficient tool for the approximation of multidimensional functions, which feature dominant directions of the periodicity. One-dimensional shift invariant spaces and tensor-product wavelets are generalized to multivariate shift invariant spaces on non-tensor-product patterns. In particular, the algebraic properties of the automorphism group are inves...
متن کاملCorrelation Kernels for Discrete Symplectic and Orthogonal Ensembles
In [41] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polyno-mials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier weights.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998