Best Fourier Approximation and Application in Efficient Blurred Signal Reconstruction

We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.     

L~p Estimates for Bi-parameter and Bilinear Fourier Integral Operators

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L~2 case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.     

Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on L₁(\mathbbR)L_{1}({\mathbb{R}}), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel.     

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation-related property and by having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

齐次超曲面上测度Fourier变换的平均衰减估计

该文主要考虑限制在齐次超曲面上的测度dμ的Fourier变换的L^q范数估计. 基于此结论, 还得到了一些有关卷积算子Tf(x, θ)=f *μ_θ(x)的混合范数不等式.     

Approximation by Partial Sums of Fourier Series and the Best Approximation of Certain Function Classes

Continuing investigations of S. A. Telyakovskii, S. B. Stechkin, A. I. Stepanets and R. M. Trigub, the author studies two questions about approximation of the class of periodic functions with a bounded -derivative by polynomials in the spaces C and L.First, the asymptotics of approximation by partial sums of Fourier series has been found without limitations for the decreasing order of . Second, the exact decreasing order of the best approximation of the class has been found. New examples are also given.     

Averaged Decay Estimates for Fourier Transforms of Measures over Curves with Nonvanishing Torsion

We study averaged decay estimates for Fourier transforms of measures when the averages are taken over space curves with non-vanishing torsion. We extend the previously known results to higher dimensions and discuss sharpness of the estimates.     

Estimates of the Best Approximation of Polynomials by Simple Partial Fractions

An asymptotics of the error of interpolation of real constants at Chebyshev nodes is obtained. Some well-known estimates of the best approximation by simple partial fractions (logarithmic derivatives of algebraic polynomials) of real constants in the closed interval [?1, 1] and complex constants in the unit disk are refined. As a consequence, new estimates of the best approximation of real polynomials on closed intervals of the real axis and of complex polynomials on arbitrary compact sets are obtained.

     

Almost Optimal Estimates for Best Approximation by Translates on a Torus

We investigate the best approximation of some classes of functions defined on a $d$-dimensional torus ${\mbox{\smallbf T}}^d$ by the transfer manifold $H_n(\varphi)=\{\xxsum_{i=1}^n b_i\varphi(\cdot -a_i)\}$ in the Hilbert space $L_2({\mbox{\smallbf T}}^d)$. For Sobolev classes of functions $\tilde{W}_2^r$ we obtain two-sided estimates of the deviation ${\rm dist}(\tilde{W}_2^r,H_n(\varphi))$. These results are applied to obtain estimates for radial basis approximation.     

Reconstruction of Hp -Functions: Best Approximation,Regularization and Optimal Error Estimates

Let U be the open unit disc of the complex plane. We explicitly construct the best pointwise approximation for determining a function in the Hardy space H ^p (U) from measured data at a countable set of points in U whenever the data is exact. A regularization scheme is given to deal with the case of nonexact data. Moreover, optimal error estimates are also studied.     

Estimates for the Approximation Numbers of One Class of Integral Operators. I

Asymptotic estimates are obtained for the approximation numbers of the Hardy integral operator with variable limits of integration.     

Estimates for the Approximation Numbers of One Class of Integral Operators. II

We obtain estimates for the Schatten–von Neumann norms of the approximation numbers of Hardy integral operator with variable limits of integration.     

On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes - The question of the representability of a continuous function on $$mathbb R^d$$ in the form of the Fourier integral of a finite Borel complex-valued measure on $$mathbb R^d$$...     

Fourier Transforms of Distributions and Hausdorff Measures

Consider a distribution whose support has Hausdorff \(h\) -measure zero. How fast can its Fourier coefficients decay to zero? If \(h\) is close to linear, we can improve on known results by a factor of about \((\log n)^{-1/2}\) .     

Marcinkiewicz-\theta-Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function . Under some conditions on we show that the maximal operator of the Marcinkiewicz- -means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where depends only on . As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz- -means of a function converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz- -means are uniformly bounded on the spaces and so they converge in norm . Some special cases of the Marcinkievicz- -summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Estimate for the Best Approximation of Summable Functions of Two Variables in Terms of Fourier Coefficients

An upper bound for the best approximation of periodic summable functions of two variables in the metric of L is obtained in terms of Fourier coefficients. Functions that can be represented by trigonometric series with coefficients satisfying a two-dimensional analog of the Boas–Telyakovskii conditions are considered.