1,720,967 research outputs found
Approximation of the Hilbert transform on the half–line
Full text linkThe paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule
Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes
No full textThe paper deals with de la Vallée Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments
On the filtered polynomial interpolation at Chebyshev nodes
No full textThe paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In previous works this has already been proved under different sufficient conditions. Here, we complete the study by stating also the necessary conditions to get it. Several numerical experiments are also given to test the theoretical results and make comparisons to Lagrange interpolation at the same nodes
Some remarks on filtered polynomial interpolation at Chebyshev nodes
Full text linkThe present paper concerns filtered de la Vallée Poussin (VP) interpolation at the Chebyshev nodes of the four kinds. This approximation model is interesting for applications because it combines the advantages of the classical Lagrange polynomial approximation (interpolation and polynomial preserving) with the ones of filtered approximation (uniform boundedness of the Lebesgue constants and reduction of the Gibbs phenomenon). Here we focus on some additional features that are useful in the applications of filtered VP interpolation. In particular, we analyze the simultaneous approximation provided by the derivatives of the VP interpolation polynomials. Moreover, we state the uniform boundedness of VP approximation operators in some Sobolev and Hölder-Zygmund spaces where several integro-differential models are uniquely and stably solvable
Filtered integration rules for finite weighted Hilbert transforms
Full text linkA product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite weighted Hilbert transform in [−1,1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov-type subspaces. Several numerical tests are provided, also comparing the rule with other formulas known in literature
Lagrange–Chebyshev Interpolation for image resizing
Full text linkImage resizing is a basic tool in image processing, and in literature, we have many methods based on different approaches, which are often specialized in only upscaling or downscaling. In this paper, independently of the (reduced or enlarged) size we aim to get, we approach the problem at a continuous scale where the underlying function representing the image is globally approximated by its Lagrange–Chebyshev I kind interpolation polynomial corresponding to suitable (tensor product) grids of first kind Chebyshev zeros. This is a well-known approximation tool widely used in many applicative fields due to the optimal behavior of the related Lebesgue constants. Here we aim to show how Lagrange–Chebyshev interpolation can be fruitfully applied also for resizing any digital image in both downscaling and upscaling at any desired size. The performance of the proposed method has been tested in terms of the standard SSIM (Structured Similarity Index Measurement) and PSNR (Peak Signal to Noise Ratio) metrics. The results indicate that, in upscaling, it is almost comparable with the classical Bicubic resizing method with slightly better metrics, but in downscaling a much higher performance has been observed in comparison with Bicubic and other recent methods too. Moreover, for all downscaling cases with an odd scale factor, we give a theoretical estimate of the MSE (Mean Squared Error) of the output image produced by our method, stating that it is certainly null (hence PSNR equals infinite and SSIM equals one) if the input image's MSE is null
Approximation of Finite Hilbert and Hadamard transforms by using equally spaced nodes
Full text linkIn the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [-1, 1] and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results
Filtered interpolation for solving Prandtl’s integro-differential equations
No full textIn order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit
An Open Image Resizing Framework for Remote Sensing Applications and Beyond
No full textImage resizing (IR) has a crucial role in remote sensing (RS), since an image’s level of detail depends on the spatial resolution of the acquisition sensor; its design limitations; and other factors such as (a) the weather conditions, (b) the lighting, and (c) the distance between the satellite platform and the ground targets. In this paper, we assessed some recent IR methods for RS applications (RSAs) by proposing a useful open framework to study, develop, and compare them. The proposed framework could manage any kind of color image and was instantiated as a Matlab package made freely available on Github. Here, we employed it to perform extensive experiments across multiple public RS image datasets and two new datasets included in the framework to evaluate, qualitatively and quantitatively, the performance of each method in terms of image quality and statistical measures
On solving some Cauchy singular integral equations by de la Vallée Poussin filtered approximation
Full text linkThe paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi–projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near–best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods
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