Approx Theory Research Articles

Existence of the solution of nonlinear fractional differential equations via new best proximity point results

AbstractIn this paper, we obtain some best proximity point results by introducing the concepts of proximal p-contractions of the first type and proximal p -contractions of the second type on partial metric spaces. Thus, some famous results in the literature such as the main result of Altun et al. (Acta Math Hung 162:393–402, 2020) and Basha (J Approx Theory 163(11):1772–1781, 2011) have been extended. Also, we provide some examples where our results are applicable and the results in Haghi et al. (Topol Appl 160:450–454, 2013) are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations via our results.

Wall Polynomials on the Real Line: A Classical Approach to OPRL Khrushchev’s Formula

The standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (Grünbaum and Velázquez in Adv Math 326:352–464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchev’s formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchev’s formula is used to rederive the characterization given in Simon (J Approx Theory 126:198–217, 2004) for the weak convergence of p_n^2mathrm{d}mu , where p_n are the orthonormal polynomials with respect to a measure mu supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchev’s formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure mu supported on an unbounded subset of the real line yields a weakly convergent sequence p_n^2mathrm{d}mu (Corollary 8.10), but there exist instances such that p_n^2mathrm{d}mu becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of p_n^2mathrm{d}mu in the unbounded case are obtained via Khrushchev’s formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchev’s formula on the real line which sheds light on its graph theoretical meaning, linked to Pólya’s recurrence theory for classical random walks.

Strong asymptotics of Jacobi-type kissing polynomials

We investigate asymptotic behaviour of polynomials satisfying varying non-Hermitian orthogonality relations where and is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials () introduced in Asheim et al. [A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Preprint, 2012 Dec 6. arXiv:1212.1293] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in Celsus and Silva [Supercritical regime for the kissing polynomials. J Approx Theory. 2020 Mar 18;225:Article ID: 105408]; Deaño [Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. J Approx Theory. 2014 Oct 1;186:33–63], and depends heavily on those works.

Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

We investigate interlacing properties of zeros of Laguerre polynomials and where and . We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp t-interval within which the zeros of two equal degree Laguerre polynomials and are interlacing for every and each is [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials and are interlacing for every and each is [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on and α, that determine the partial or full interlacing of the zeros of and the zeros of . We also prove that partial interlacing holds between the zeros of and when and . Numerical illustrations of interlacing and its breakdown are provided.

Strong Partially Greedy Bases and Lebesgue-Type Inequalities

In this paper, we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in Dilworth et al. (Constr Approx 19:575–597, 2003) for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in Albiac and Ansorena (Rev Matem Compl 30(1):13–24, 2017), Albiac and Wojtaszczyk (J Approx Theory 138:65–86, 2006).

Uniqueness of minimal projections in smooth expanded matrix spaces

Let us consider the space M(n, m) of all real or complex matrices on n rows and m columns. In 2000 Lesław Skrzypek proved the uniqueness of minimal projection of this space onto its subspace M(n,1)+M(1,m) which consists of all sums of matrices with constant rows and matrices with constant columns. We generalize this result using some new methods proved by Lewicki and Skrzypek (J Approx Theory 148:71–91, 2007). Let S be a space of all functions from Xtimes Y times Z into {mathbb {R}} or {mathbb {C}}, where X, Y, Z are finite sets. It could be interpreted as a space of three-dimensional matrices M(n, m, r). Let T be a subspace of S consisting of all sums of functions which depend on one variable. Let S be equipped with a smooth norm Vert .Vert . We show that there exists the unique minimal projection of S onto its subspace T.

On a joint approximation question in $$H^p$$ spaces

We consider Mergelyan sets and Farrell sets for $$H^p$$ $$(1\le p < \infty )$$ spaces in the unit disc for both the weak topology and the norm topology, and give a short proof of a theorem of Perez-Gonzalez which answers a question proposed by Rubel and Stray (J Approx Theory 37:44–50, 1983).

A Fractalization of Rational Trigonometric Functions

In Navascues (Z Anal Anwend 25(2):401–418, 2005) and Viswanathan and Chand (J Approx Theory 185:31–50, 2014), new approximation classes of self-referential functions are introduced as fractal versions of the classes of polynomials and rational functions. As a sequel, in the present article, we define a new approximation class consisting of self-referential functions, referred to as the fractal rational trigonometric functions. We establish Weierstrass type approximation theorems for this class and prove the existence of a best fractal rational trigonometric approximant to a real-valued continuous function on a compact interval. Furthermore, we provide an upper bound for the smallest error in approximating a prescribed continuous function by a fractal rational trigonometric function. This extemporizes an analogous result in the context of fractal rational function appeared in Viswanathan and Chand (J Approx Theory 185:31–50, 2014) and followed in the setting of Bernstein fractal rational functions in Vijender (Fractals 26(4):1850045, 2018). The last part of the article aims to clarify and correct the mathematical errors in some results on the Bernstein $$\alpha $$-fractal functions appeared recently in the literature (Vijender in Acta Appl Math 159:11–27, 2019, Vijender in Fractals 26(4):1850045, 2018, Vijender in Mediterr J Math 211:1–16, 2018).

Central orderings for the Newton interpolation formula

The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017. https://doi.org/10.1016/j.jat.2017.07.005; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included.

Approximation of Integrable Functions by Wavelet Expansions

Walter (J Approx Theory 80:108–118, 1995), Xiehua (Approx Theory Appl 14(1):81–90, 1998) and Lal and Kumar (Lobachevskii J Math 34(2):163–172, 2013) established results on pointwise and uniform convergence of wavelet expansions. Working in this direction new more general theorems on degree of pointwise approximation by such expansions have been proved.

A new characterization of convergent multivariate subdivision schemes with nonnegative masks

We present in this paper new necessary and sufficient conditions for convergence of multivariate subdivision schemes with nonnegative finite masks, which simplify the assertion obtained by Wang in (J Approx Theory 113:207–220, 2001). Moreover, we construct an example, which shows that the convergence behaviors for univariate subdivision schemes and multivariate ones are essentially different.

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

We study the Hamiltonian of a two-dimensional log-gas with a confining potential V satisfying the weak growth assumption—V is of the same order as $$2\log \Vert x\Vert $$ near infinity—considered by Hardy and Kuijlaars [J Approx Theory 170:44–58, 2013]. We prove an asymptotic expansion, as the number n of points goes to infinity, for the minimum of this Hamiltonian using the gamma-convergence method of Sandier and Serfaty [Ann Probab 43(4):2026–2083, 2015]. We show that the asymptotic expansion as $$n\rightarrow +\infty $$ of the minimal logarithmic energy of n points on the unit sphere in $$\mathbb {R}^3$$ has a term of order n, thus proving a long-standing conjecture of Rakhmanov et al. [Math Res Lett 1:647–662, 1994]. Finally, we prove the equivalence between the conjecture of Brauchart Brauchart, Hardin and Saff [Contemp. Math., 578:31–61, 2012] about the value of this term and the conjecture of Sandier and Serfaty [Commun Math Phys. 313(3):635–743, 2012] about the minimality of the triangular lattice for a “renormalized energy” W among configurations of fixed asymptotic density.

Convergence for iterative methods on Banach spaces of a convergence structure with applications to fractional calculus

We present a semilocal convergence for some iterative methods on a Banach space with a convergence structure to locate zeros of operators which are not necessarily Frechet-differentiable as in earlier studies such as (Argyros in J Approx Theory Appl 9(1):1–10, 1993; Argyros in Southwest J Pure Appl Math 1:32–38, 1995; Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992). This way we expand the applicability of these methods. If the operator involved is Frechet-differentiable one approach leads to more precise error estimates on the distances involved than before (Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992) and under the same hypotheses. Special cases are presented and some examples from fractional calculus.

Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials $$\small {}_{A}\text {F}_{B}\left[ \begin{array}{c} -n,a_2,\ldots ,a_A \\ b_1,b_2,\ldots ,b_B \end{array} ; \begin{array}{cc} z \end{array}\right] $$ , using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it.

Lorentz Spaces and Embeddings Induced by Almost Greedy Bases in Banach Spaces

The aim of this paper is to undertake a systematic qualitative study of the built-in symmetry of almost greedy bases in Banach spaces. More specifically, by refining the techniques that Wojtaszczyk used in J Approx Theory 107(2), 293–314 2000 for quasi-greedy bases in Hilbert spaces, we show that an almost greedy basis in a Banach space X naturally induces embeddings that allow sandwiching X between two symmetric sequence spaces. Using classical interpolation techniques in combination with duality, we also explore what we label as interpolation of greedy bases. It is then proved that the only almost greedy basis shared by any two \(\ell _{p}\) spaces is equivalent to the standard unit vector basis and that there is no basis which is simultaneously (normalized and) greedy in two different \(L_{p}\) spaces. As a by-product of our work, we obtain a new characterization of greedy bases in Banach spaces in terms of bounded linear operators.

Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

In the paper we investigate the continuity properties of the mapping \({\Phi}\) which sends any non-empty compact connected hv-convex planar set K to the associated generalized conic function fK. The function fK measures the average taxicab distance of the points in the plane from the focal set K by integration. The main area of applications is geometric tomography because fK involves the coordinate X-rays’ information as second order partial derivatives (Nagy and Vincze, J Approx Theory 164: 371–390, 2012). We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the L1-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that \({\Phi^{-1}}\) is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if fL is close to fK then L must be close to an element \({K^\prime}\) such that \({f_{K}=f_{K^\prime}}\). Therefore K and \({K^\prime}\) have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies K for which fK is a point of lower semi-continuity for \({\Phi^{-1}}\).

Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme

This paper will demonstrate how European and American option prices can be computed under the jump-diffusion model using the radial basis function (RBF) interpolation scheme. The RBF interpolation scheme is demonstrated by solving an option pricing formula, a one-dimensional partial integro-differential equation (PIDE). We select the cubic spline radial basis function and adopt a simple numerical algorithm (Briani et al. in Calcolo 44:33–57, 2007) to establish a finite computational range for the improper integral of the PIDE. This algorithm reduces the truncation error of approximating the improper integral. As a result, we are able to achieve a higher approximation accuracy of the integral with the application of any quadrature. Moreover, we a numerical technique termed cubic spline factorisation (Bos and Salkauskas in J Approx Theory 51:81–88, 1987) to solve the inversion of an ill-conditioned RBF interpolant, which is a well-known research problem in the RBF field. Finally, our numerical experiments show that in the European case, our RBF-interpolation solution is second-order accurate for spatial variables, while in the American case, it is second-order accurate for spatial variables and first-order accurate for time variables.

A connection between Chidume iteration and the contraction-proximal point algorithm

In this short note we comment on some recent results on the convergence of two type of iterative algorithms for finding a zero of a maximal monotone operator. The first one is based on a proximal approach and was investigated by many authors, see for example (Boikanyo and Morosanu in Nonlinear Anal TMA 74:544–555, ;2011 Kamimura and Takahashi in J Approx Theory 106:226–240, 2000, Marino and Xu in Comm Pure Appl Anal 3:791–808, 2004, Wang and Cui in J Global Optim 54:485–491, 2012, Yao and Noor in J Comput Appl Math 217:46–55, 2008) and the references therein. The second is based on a gradient approach and was essentially developed by Chidume et al. see for instance (Fixed Point Theory 2013, Chidume and Djitte in Abst Appl Anal doi:10.1155/2012/6813482012, Chidume et al. in J Nonlinear Conv Anal (to appear) 2013, Ofoedu and Zegeye in J Math Anal Appl 372:68–76, 2010) and the references therein. Then, relying on the Yosida approximate, we compare and make a link between these two approaches.

Error estimates for the modified truncations of approximate approximation with Gaussian kernels

We introduce a type of modified truncations of approximate approximation with Gaussian kernels, which can be applied to approximate functions on compact intervals. Our results improve the related results of Chen and Cao (Appl Math Comput 217:725---734, 2010), Muller and Varnhorn (J Approx Theory 145:171---181, 2007). Also, we construct approximate approximation operators in multivariate cases and obtain the estimates of approximation rate.

A supplement to a regularization method for the proximal point algorithm

The purpose of this paper is to show that the iterative scheme recently studied by Xu (J Glob Optim 36(1):115---125, 2006) is the same as the one studied by Kamimura and Takahashi (J Approx Theory 106(2):226---240, 2000) and to give a supplement to these results. With the new technique proposed by Mainge (Comput Math Appl 59(1):74---79, 2010), we show that the convergence of the iterative scheme is established under another assumption. It is noted that if the computation error is zero or the approximate computation is exact, our new result is a genuine generalization of Xu's result and Kamimura---Takahashi's result.