Geodesic
Efficient computation of Favard constants and their connection to Euler polynomials and numbers
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1921-1942.

Voir la notice de l'article provenant de la source Math-Net.Ru

We discuss problems of calculating the Favard constants, which are often used in approximation theory and their connection to Euler numbers and polynomials. Simple effective recurrence formulas for computation of the Favard constants are found. The application of the results to one problem of extremal functional interpolation allowing the solution to be expressed in an explicit form is demonstrated.
Keywords: Euler numbers, recurrence formulas, approximation theory.
Mots-clés : Favard constants, Euler polynomials 
@article{SEMR_2020_17_a142,
     author = {Yu. S. Volkov},
     title = {Efficient computation of {Favard} constants and their connection to {Euler} polynomials and numbers},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1921--1942},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a142/}
}
TY  - JOUR
AU  - Yu. S. Volkov
TI  - Efficient computation of Favard constants and their connection to Euler polynomials and numbers
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 1921
EP  - 1942
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a142/
LA  - en
ID  - SEMR_2020_17_a142
ER  - 
%0 Journal Article
%A Yu. S. Volkov
%T Efficient computation of Favard constants and their connection to Euler polynomials and numbers
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 1921-1942
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a142/
%G en
%F SEMR_2020_17_a142
Yu. S. Volkov. Efficient computation of Favard constants and their connection to Euler polynomials and numbers. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1921-1942. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a142/

[1] N. Korneichuk, Exact constants in approximation theory, Cambridge University Press, Cambridge, 1991 | MR | Zbl

[2] V.M. Tikhomirov, “Approximation theory”, Analysis, v. II, Encycl. Math. Sci., 14, Convex analysis and approximation theory, Springer, Berlin–Heidelberg, 1990, 93–255 | Zbl

[3] R.A. DeVore, G.G. Lorentz, Constructive approximation, Springer, Berlin, 1993 | MR | Zbl

[4] I.J. Schoenberg, “Cardinal interpolation and spline functions”, J. Approx. Theory, 2:2 (1969), 167–206 | DOI | MR | Zbl

[5] I.J. Schoenberg, “Cardinal interpolation and spline functions: II interpolation of data of power growth”, J. Approx. Theory, 6:4 (1972), 404–420 | DOI | MR | Zbl

[6] A.S. Cavaretta Jr., “On cardinal perfect splines of least sup-norm on the real axis”, J. Approx. Theory, 8:4 (1973), 285–303 | DOI | MR | Zbl

[7] G. Glaeser, “Prolongement extremal de fonctions différentiables”, J. Approx. Theory, 8:3 (1973), 249–261 | DOI | MR | Zbl

[8] J. Favard, “Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynomes trigonométriques”, Bull. Sci. Math., II. Ser., 61 (1937), 209–224 ; 243–256 | MR | Zbl

[9] N.I. Akhiezer, M.G. Krein, “On the best approximation of periodic functions”, Dokl. Akad. Nauk SSSR, 15:3 (1937), 107–111 | Zbl

[10] Yu.N. Subbotin, “Norms of interpolational splines of odd degree in the spaces $W_2^k$”, Math. Notes, 44:6 (1988), 958–962 | DOI | MR | Zbl

[11] V.V. Zhuk, S.Yu. Pimenov, “On norms of Akhiezer-Krein-Favard sums”, Vestnik St. Peterburg Univ., Prikl. Math. Inf. Pros. Upravl., 2006:4, 37–47 | MR

[12] A.N. Kolmogorov, “On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval”, Amer. Math. Soc. Transl., 1949, no. 2 | MR

[13] Yu.S. Volkov, Yu.N. Subbotin, “Fifty years of Schoenberg's problem on the convergence of spline interpolation”, Proc. Steklov Inst. Math., 288, suppl. 1 (2015), S222–S237 | DOI | MR | Zbl

[14] Yu.S. Volkov, “On one problem of extremal functional interpolation and Favard constants”, Dokl. Math., 102:3 (2020) | DOI

[15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher transcendental functions, v. 1, McGraw-Hill, New York, 1953 | Zbl

[16] A.P. Prudnikov, A.Yu. Brychkov, O.L. Marichev, Integrals and series, v. 1, Elementary functions, Gordon Breach Science Publishers, New York etc., 1986 | MR | Zbl

[17] K. Dilcher, “Bernoulli and Euler polynomials”, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, 2010, 587–599 | MR

[18] M. Abramovitz, I.A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards, John Wiley Sons, New York etc, 1972 | MR | Zbl

[19] A.I. Stepanets, Methods of Approximation Theory, VSP, Utrecht, 2005 | MR | Zbl

[20] V.L. Miroshnichenko, “On exact estimates of approximation by interpolation splines”, Siberian conf. «Methods of spline functions», Abstracts, Sobolev Inst. Math., 2001, 55–56

[21] N.E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924 | Zbl

[22] A.F. Horadam, “Genocchi polynomials”, Applications of Fibonacci numbers, v. 4, Kluwer, Dordrecht, 1990, 145–166 | MR | Zbl

[23] M.K. Kerimov, “The Rayleigh function: Theory and methods for its calculation”, Comput. Math. Math. Phys., 39:12 (1999), 1883–1925 | MR | Zbl

[24] S.G. Gocheva-Ilieva, I.H. Feschiev, “New recursive representations for the Favard constants with application to multiple singular integrals and summation of series”, Abstr. Appl. Anal., 2013, 523618 | MR | Zbl

[25] A.G. Kuznetsov, I.M. Pak, A.E. Postnikov, “Increasing trees and alternating permutations”, Russ. Math. Surv., 49:6 (1994), 79–114 | DOI | MR | Zbl

[26] R. Adin, Y. Roichman, “Standard Young tableaux”, Handbook of Enumerative Combinatorics, ed. M. Bóna, CRC Press, Boca Raton, 2015, 895–974 | MR | Zbl

[27] V.I. Arnold, “Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics”, Duke Math. J., 63:2 (1991), 537–555 | DOI | MR | Zbl

[28] https://oeis.org/

[29] L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Company, Dordrecht, 1974 | MR | Zbl

[30] I.J. Schoenberg, Cardinal spline interpolation, SIAM, Philadelphia, 1973 | MR | Zbl

[31] G.V. Demidenko, V.L. Vaskevich (eds.), Selected Works of S.L. Sobolev, v. I, Equations of mathematical physics, computational mathematics, and cubature formulas, Springer, New York, 2006 | MR | Zbl

[32] S.B. Stechkin, Yu.N. Subbotin, Splines in numerical mathematics, Nauka, M., 1976 | MR | Zbl

[33] Yu.N. Subbotin, “Interpolation by functions with $n$th derivative of minimum norm”, Proc. Steklov Inst. Math., 88 (1967), 31–63 | Zbl

[34] Yu.N. Subbotin, “Approximation by spline functions and estimates of diameters”, Proc. Steklov Inst. Math., 109 (1971), 39–67 | MR | Zbl

[35] Yu.N. Subbotin, “Extremal problems of functional interpolation and interpolation-in-the-mean splines”, Proc. Steklov Inst. Math., 138 (1977), 127–185 | MR | Zbl

[36] E.V. Mishchenko, “Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials”, Sib. Math. J., 51:4 (2010), 660–666 | DOI | MR | Zbl

[37] E.V. Mishchenko, “Construction of polynomials of a special kind and examples of their application in the theory of power series and in the wavelet theory”, Russian Math., 55:1 (2011), 19–32 | DOI | MR | Zbl

[38] Yu.N. Subbotin, S.I. Novikov, V.T. Shevaldin, “Extremal function interpolation and splines”, Tr. Inst. Mat. Mekh. UrO RAN, 24, no. 3, 2018, 200–225 | MR

[39] Yu.N. Subbotin, “On the relations between finite differences and the corresponding derivatives”, Proc. Steklov Inst. Math., 78 (1965), 24–42 | MR | Zbl

[40] J. Favard, “Sur l'interpolation”, J. Math. Pures Appl., 19:3 (1940), 281–306 | MR | Zbl

[41] C. de Boor, “A smooth and local interpolant with small $k$-th derivative”, Numeri. Solut. Bound. Value Probl. ordin. differ. Equat., Proc. Symp. (Baltimore, 1974), Academic Press, New York, 1975, 177–197 | MR | Zbl

[42] C. de Boor, How small can one make the derivatives of an interpolating function?, J. Approx. Theory, 13:2 (1975), 105–116 | DOI | MR | Zbl

[43] C. de Boor, “On 'best' interpolation”, J. Approx. Theory, 16:1 (1976), 28–42 | DOI | MR | Zbl

[44] H.-O. Kreiss, “Difference approximations for singular perturbation problems”, Numer. Solut. Bound. Value Probl. ordin. differ. Equat., Proc. Symp. (Baltimore, 1974), Academic Press, New York, 1975, 199–211 | MR | Zbl

[45] C. de Boor, “The exact condition of the B-spline basis may be hard to determine”, J. Approx. Theory, 60:3 (1990), 344–359 | DOI | MR | Zbl

[46] K. Scherer, A.Yu. Shadrin, “New upper bound for the B-spline basis condition number II. A proof of de Boor's $2^k$-conjecture”, J. Approx. Theory, 99:2 (1999), 217–229 | DOI | MR | Zbl

[47] T. Lyche, “A note on the condition numbers of the B-spline bases”, J. Approx. Theory, 22:3 (1978), 202–205 | DOI | MR | Zbl

[48] S.I. Novikov, V.T. Shevaldin, “On the connection between the second divided difference and the second derivative”, Tr. Inst. Mat. Mekh. UrO RAN, 26, no. 2, 2020, 216–224 | MR

[49] J. Liu, H. Pan, Y. Zhang, “On the integral of the product of the Appell polynomials”, Integral Transforms Spec. Funct., 25:9 (2014), 680–685 | DOI | MR | Zbl

[50] F. Qi, “A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers”, J. Comput. Appl. Math., 351 (2019), 1–5 | MR | Zbl