# Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

## Abstract

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, *q* _{1}, and *q* _{2} for which the known orthogonal systems are everywhere dense in asymmetric spaces *L* _{(α,β);q} ([0, 1]) are found.

**Theorem.** Let α, β, *q* _{1}, *q* _{2} ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces *L* _{(α,β);q} ([0, 1]) if and only if either *max*{α, β, *q* _{1}, *q* _{2}} < + ∞ or *max* {α, β} < +∞, *q* _{1} = *q* _{2} = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system.

## Keywords

Closed Interval Trigonometric Polynomial Orthogonal System Linear Hull Weierstrass Theorem## Preview

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## References

- 1.A. R. Alimov, “A number of connected components of sun’s complement,”
*East J. Approx.*,**1**, No. 4, 419–429 (1995).MATHMathSciNetGoogle Scholar - 2.A. R. Alimov, “Chebyshev sets’ components,”
*East J. Approx.*,**2**, No. 2, 215–232 (1996).MATHMathSciNetGoogle Scholar - 3.A. R. Alimov, “Chebyshev compact sets on the plane,”
*Tr. Mat. Inst. Ross. Akad. Nauk*,**219**, 3–22 (1997).Google Scholar - 4.A. R. Alimov, “Chebyshev sets in linear spaces with asymmetrical sphere,” in:
*Trudy Saratov Winter School January 30–February 4, 1994 (to the memory of Professor A. A. Privalov)*[in Russian],*Mezhvysovskii Sbornik Nauchnykh Trudov. Part 2*, Izd. Saratov Univ., (1995), pp. 91–93.Google Scholar - 5.A. R. Alimov,
*Approximative Properties of Sets in Linear Spaces with Asymmetrical Sphere*[in Russian], Thesis, MGU, Moscow (1997).Google Scholar - 6.A. S. Andreev, V. A. Popov, and B. Sendov, “Jackson-type theorems for best one-sided approximations by trigonometric polynomials and splines,”
*Mat. Zametki*,**26**, 791–804 (1979).MATHMathSciNetGoogle Scholar - 7.A. S. Andreev, V. A. Popov, and B. Sendov, “Jackson-type theorems for one-side polynomial and spline approximation,”
*Bulgar Acad. Sci*,**30**, 1533–1536 (1977).MATHMathSciNetGoogle Scholar - 8.V. V. Arestov, “On the inequality of different metrics for trigonometric polynomials,”
*Mat. Zametki*,**27**, No. 4, 539–546 (1980).MATHMathSciNetGoogle Scholar - 9.V. F. Babenko, “Asymmetrical approximations in spaces of summable functions,”
*Ukr. Mat. Zh.*,**34**, No. 4, 409–416 (1982).MATHMathSciNetGoogle Scholar - 10.V. F. Babenko, “Asymmetrical extremal problems of approximation theory,”
*Dokl. Akad. Nauk SSSR*,**269**, No. 3, 521–524 (1983).MATHMathSciNetGoogle Scholar - 11.V. F. Babenko, “Inequalities for permutations of differentiable functions, problems of approximation and approximate integration,”
*Dokl. Akad. Nauk SSSR*,**272**, No. 5, 1038–1041 (1983).MATHMathSciNetGoogle Scholar - 12.V. F. Babenko, “Asymmetrical approximations and inequalities for permutations in extremal problems of approximation theory,”
*Tr. Mat. Inst. Akad. Nauk SSSR*,**180**, 33–35 (1987).MathSciNetGoogle Scholar - 13.V. F. Babenko and M. B. Vakarchuk, “On inequalities of Kolmogorov-Hormander type for functions bounded on the discrete grid,”
*Ukr. Mat. Zh.*,**49**, No. 7, 988–992 (1997).MATHMathSciNetGoogle Scholar - 14.V. F. Babenko and V. A. Kofanov, “Asymmetrical approximations of classes of differentiable functions by algebraic polynomials in the mean,”
*Anal. Math.*,**14**, No. 3, 149–217 (1988).MathSciNetGoogle Scholar - 15.V. F. Babenko and A. A. Ligun, “Order of best one-sided approximations by polynomials and splines in the metric
*L*_{p},”*Mat. Zametki*,**19**, 323–329 (1976).MATHMathSciNetGoogle Scholar - 16.R. Bojanic and R. De Vore, “On polynomials of best one-sided approximation,”
*Enseign. Math.*,**12**, 139–164 (1966).MATHGoogle Scholar - 17.A. Bronsted, “Convex sets and Chebyshev sets,”
*Math. Scand.*,**17**, 5–16 (1965).MathSciNetGoogle Scholar - 18.A. Bronsted, “Convex sets and Chebyshev sets. 2,”
*Math. Scand.*,**18**, 5–15 (1966).MathSciNetGoogle Scholar - 19.E. P. Dolzhenko and E. A. Sevast’yanov, “On approximation of functions in the Hausdorff metric by means of piecewise-monotone (in particular, rational) functions,”
*Mat. Sb.*,**101**, 508–541 (1976).MATHMathSciNetGoogle Scholar - 20.E. P. Dolzhenko and E. A. Sevast’yanov, “Sign-sensitive approximations. The space of sign-sensitive weights. Rigidity and freedom of the system,”
*Dokl. Ross. Akad. Nauk*,**332**, No. 6, 686–689 (1993).MATHGoogle Scholar - 21.E. P. Dolzhenko and E. A. Sevast’yanov, “Sign-sensitive approximations. Problems of uniqueness and stability,”
*Dokl. Ross. Akad. Nauk*,**333**, No. 1, 5–7 (1993).MATHGoogle Scholar - 22.E. P. Dolzhenko and E. A. Sevast’yanov, “Approximation with the sign-sensitive weight,”
*Izv. Ross. Akad. Nauk*,**62**, No. 6, 59–102 (1998).MATHMathSciNetGoogle Scholar - 23.E. P. Dolzhenko and E. A. Sevast’yanov, “On the definition of Chebyshev grass-snakes,”
*Vestn. MGU, Ser. 1, Mat., Mekh.*, No. 3, 49–59 (1994).Google Scholar - 24.G. Freud, “Uber einseitige approximation durch polynome. I,”
*Acta Sci. Math.*,**16**, 12–18 (1955)MATHMathSciNetGoogle Scholar - 25.T. Ganelius, “On one-sided approximation by trigonometrical polynomials,”
*Math. Scand.*,**4**, 247–258 (1956).MATHMathSciNetGoogle Scholar - 26.V. N. Garbushin, “New inequalities for derivatives and their applications,” in:
*Int. Conf. on Functional Spaces, Approximation Theory and Nonlinear Analysis Dedicated to the 90th Birthday of Academician S. M. Nikolskii*[in Russian],*Moscow, April 27–May 3*, 90–91, Moscow (1995).Google Scholar - 27.L. Hörmander, “A new proof and generalization of an inequality of Bohr,”
*Math. Scand.*,**2**, 33–45 (1954).MATHMathSciNetGoogle Scholar - 28.B. S. Kashin and A. A. Saakyan,
*Orthogonal Series*[in Russian], AFTs, Moscow (1999).MATHGoogle Scholar - 29.A. N. Kolmogorov, “On inequalities between upper faces of sequential derivatives of functions on an infinite interval,”
*Uch. Zap. Mosk. Univ., Mat.*,**30**, No. 3, 3–13 (1939).Google Scholar - 30.N. P. Korneichuk,
*Exact Constants in Approximation Theory*[in Russian], Nauka, Moscow (1987).MATHGoogle Scholar - 31.N. P. Korneichuk, A. A. Ligun, and V. G. Doronin,
*Approximation with Constraints*[in Russian], Naukova Dumka, Kiev (1982).MATHGoogle Scholar - 32.A. I. Kozko, “Analogs of Jackson-Nikolskii inequalities for trigonometric polynomials in spaces with asymmetrical norm,”
*Mat. Zametki*,**61**, No. 5, 687–699 (1997).MATHMathSciNetGoogle Scholar - 33.A. I. Kozko, “On Jackson-Nikolskii inequalities for trigonometric polynomials in spaces with asymmetrical norms,”
*East J. Approx.*,**2**, No. 2, 177–186 (1996).MATHMathSciNetGoogle Scholar - 34.A. I. Kozko, “Multidimensional inequalities of different metrics in spaces with asymmetrical norm,”
*Mat. Sb.*,**189**, No. 9, 85–106 (1998).MATHMathSciNetGoogle Scholar - 35.A. I. Kozko, “Inequalities with fractional derivatives for trigonometric polynomials in spaces with asymmetric norm,”
*Izv. Ross. Akad. Nauk, Ser. Mat.*,**62**, No. 6 (1998)Google Scholar - 36.A. I. Kozko, “On the order of best approximation in spaces with asymmetrical norm on the classes of functions with bounded
*r*th derivative,”*Izv. Ross. Akad. Nauk, Ser. Mat.*(2001).Google Scholar - 37.M. G. Krein, “
*L*-problem of moments in the abstract linear normed space,” in: N. I. Akhiezer and M. G. Krein,*Some Questions in the Theory of Moments*[in Russian], GONTI, Khar’kov (1938).Google Scholar - 38.G. Lorentz, M. V. Golitsihek and Y. Makovoz, “Constructive approximation: Advanced problems,”
*Comp. Stud. Math.*,**304**(1996).Google Scholar - 39.N. N. Luzin,
*Integrals*[in Russian], Moscow (1948).Google Scholar - 40.G. G. Magaril-Il’yaev and V. M. Tikhomirov,
*Convex Analysis and Its Applications*[in Russian], Editorial URSS, Moscow (2000).Google Scholar - 41.
- 42.O. V. Polyakov, “Asymmetrical approximations of a class by the class assigned with the help of a linear differential operator,”
*Ukr. Mat. Zh.*,**42**, No. 8, 1083–1088 (1990).MATHMathSciNetCrossRefGoogle Scholar - 43.O. V. Polyakov, “On approximation by generalized splines of certain classes of differentiable functions,”
*Ukr. Mat. Zh.*,**49**, No. 7, 951–957 (1997).MATHMathSciNetGoogle Scholar - 44.A.-R. K. Ramazanov, “On direct and inverse theorems of approximation in the metric of sign-sensitive weight,”
*Anal. Math.*,**21**, No. 3, 191–212 (1995).MATHMathSciNetCrossRefGoogle Scholar - 45.A.-R. K. Ramazanov, “Orthogonal polynomials with sign-sensitive weight,”
*Mat. Zametki*,**59**, No. 5, 737–752 (1996).MATHMathSciNetGoogle Scholar - 46.A.-R. K. Ramazanov, “Rational approximation with sign-sensitive weight,”
*Mat. Zametki*,**60**, No. 5, 737–752 (1996).Google Scholar - 47.B. Sendov and V. A. Popov,
*Averaged Smoothness Modules*, Bulgar Akad. Nauk, Sophia (1983).Google Scholar - 48.E. A. Sevast’yanov, “On the Haar problem for sign-sensitive approximations,”
*Mat. Sb.*,**188**, No. 2, 95–128 (1997).MathSciNetGoogle Scholar - 49.A. Yu. Shadrin,
*One-Sided and Monotone Approximations*[in Russian], Thesis, Moscow (1986).Google Scholar - 50.A. A. Shumeiko, “Asymmetrical spline approximations,” in:
*Int. Conf. on Functional Spaces, Approximation Theory and Nonlinear Analysis Dedicated to the 90th Birthday of Academician S. M. Nikolskii*[in Russian],*Moscow, April 27–May 3*, Moscow (1995).Google Scholar - 51.V. M. Tikhomirov, “Theory on extremum and extremal problems of classical analysis,” in:
*Progress in Science and Technology, Series on Contemporary Problems in Mathematics and Applications*[in Russian], VINITI, Moscow,**65**, 188–258 (1999).Google Scholar