### Constructive function theory, - interpolation and approximation quadratures

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Finally, numerical examples and comparison with other available methods are presented to illustrate the usefullness of proposed method. We construct here a class of interpolating splines of degree Error estimates for this spline is also represented. This gives the above recurrence formula 3. Thus, the proof is completed. In this section, error estimates for the above interpolatory twelfth spline is considered.

The inequality 6 provides a correction of inequalities 5 , 10 , In this section, we illustrate the numerical technique discussed in the previous section by the following problems, in order to illustrate the comparative performance of the proposed spline method over other existing spline methods. In Table 2 , maximum errors are reported corresponding to the present spline method and the spline method in El Tarazi and Karaballi for various values of h.

Also, the exact and the numerical solutions are plotted in Fig. It has been observed that our method is more efficient. Consider the following function Phythian and Williams :. The maximum absolute errors are tabulated in Table 6 for various values of h and compared with Anwar and El-Tarazi The proposed numerical solution gives almost overlapping behavior with the corresponding exact solution values.

The numerical results are tabulated in Table 8. It is clear from the Tables 1 , 2 , 3 , 4 , 5 , 6 , 7 and 8 that our methods are better than the other existing methods. In this paper, the existence and uniqueness of a twelfth degree spline are derived and in which we have obtained a direct simple formula. For which, error estimates and numerical examples are presented. On the basis of the examples, the proposed method yields much better results than the other methods.

Also, a mistake is corrected in the literature that occurred in error bounds. Commun Appl Numer Methods — Birkhoff G, Priver A Hermite interpolation errors for derivatives. J Math Phys — Numer Math — J Approx Theory — Phythian J, Williams R Direct cubic spline approximation to integrals. Int J Numer Methods Eng — Int J Math Anal — Varma A, Howell G Best error bounds for derivatives in two point Birkhoff interpolation problems.

Download references. POM is one of the authors of the original idea, mathematical background and all numerical simulations. He proved the Existence and Uniqueness of the proposed method and presented the first two examples. On Certain Extremal Problems for Polynomials. A quadrature formula with zeros of Bessel functions as nodes. The summation formulae of Poisson, Plana, Euler-Maclaurin and their relationship.

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Anal Appl. A quadrature formula for entire functions of exponential type. Entire functions of exponential type belonging to L p R. Characterization of Entire Functions via Quadrature. Extension of a theorem of J. Grace to transcendental entire functions. Case of Non-periodic Functions. On extremal properties of the derivatives of polynomials and rational functions. Characterization of functions in terms of rate of convergence of a quadrature process.

Bulletin of the Polish Acad. Characterization of the Speed of Convergence of the Trapezoidal Rule. Converse Theorems in the Theory of Approximate Integration. L p inequalities for entire functions of exponential type. Quadrature formulae and functions of exponential type. On Polynomials with Curved Majorants. Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant.

Algebra Applics. L p Inequalities for Polynomials. Sur le theoreme de Koebe-Bieberbach. On a problem related to the conjecture of Sendov about the critical points of a polynomial. Representation and Approximation of Functions via 0 , 2 -Interpolation. Mariae Curie-Sklodowska Sect. Sur une formule de quadrature pour des functions entieres. On Polynomials with a Prescribed Zero. Une formule de quadrature pour les functions entieres de type exponential.

Inequalities for Polynomials with Two Equal Coefficients. New inequalities for polynomials. Pointwise Bounded Families of Holomorphic Functions. On pairs of coefficients of bounded polynomials.

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An inequality for asymmetric entire functions. On the zeros of rational functions arising from certain determinants. Research problems in Function Theory.

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## Publications of Yuan Xu

On an inequality of S. On the Hadamard products of schlicht functions. On a problem of Turan about polynomials. Coefficient regions for univalent trinomials. On the derivative of a polynomial. Illinois J. An extremal problem for polynomials with a prescribed zero. On some classes of univalent polynomials.

On a phenomenon concerning 0 , 2 -interpolation. On Simultaneous Chebyshev Approximation. Inequalities for polynomials on the unit interval. On rational approximation on the positive real axis. Location of the Zeros of Polynomials with a Prescribed Norm. Some inequalities for polynomials. Some inequalities for polynomials with a prescribed zero. Differential inequalities and local valency. Pacific J. Location of the zeros of polynomials with a prescribed norm. Inequalities for polynomials with a prescribed zero. On a property of rational functions.

On the zeros of a polynomial and its derivative. On a problem of Turan about polynomials with curved majorants. Zeros of linear combinations of polynomials. The distinct zeros of the product of a polynomial and its successive derivatives. A bound for the moduli of the zeros of polynomials. Carlson's theorem for functions of exponential type. Analyse Math.

## Prof. Giuseppe Mastroianni

Remarque sur la moyenne arithmetique de fonctions univalentes convexes. Functions of exponential type not vanishing in a half plane and related polynomials. Functions of exponential type. Some inequalities concerning functions of exponential type. Remarque sur la moyenne geometrique de fonctions univalentes convexes.

Paris Ser. L'influence des zeros sur la borne dans le theoreme de Markoff pour les polynomes sur l'intervalle unite. On the Enestrom-Kakeya theorem. Tohoku Math. On the monotonicity of certain functionals in the theory of analytic functions. Polynomials with some prescribed zeros. Univalent Polynomials of Small Degree.

On the location of zeros of polynomials. Remarks on Schwarz's lemma. A property of entire functions of exponential type. The second mean values of entire functions. On asymmetric entire functions. Extremal problems and polynomials of least deviation. Scripta Math. The Ritt order of the derivative of an entire function. Indian J. Interpolation of entire functions. Extremal properties of the successive derivatives of polynomials and rational functions. Some inequalities for polynomials and related entire functions.

Inequalities for monotonic entire functions. Michigan Math. Inequalities Concerning Polynomials and Trigonometric Polynomials. An inequality for trigonometric polynomials. Rational Mech. Some inequalities for polynomials and entire functions. The influence of coefficients on the zeros of polynomials.