Best approximation in Banach spaces; duality theorems for best approximation; conditions of existance and uniqueness of best approximation; geometry of a Banach sphere; Chebyshev systems; Chebyshev centre; approximation in the spaces of continuous and summables functions; approximation of functions of several variables; superpositions of functions.
Subject:
The following results are obtained: on simultaneous approximation of a function with its derivatives; on criterion of best approximation; duality theorems of convex approximation; conditions of existance and uniqueness of best approximation in general Banach spaces, in Frechet spaces and in spaces of continued and summable functions; on conditions of existance and uniqueness of Chebyshev centre; on the best approximation of functions of several variables by linear superpositions; on optimization of building process with random parameters.
Biography
Graduated from Lwow University in 1956 and post-graduate course in Moscow Mathem. Steklov Institute (MIAN) in 1959 (department of theory of function and fuctional analysis). Ph.D. thesis was defended in 1960. D.Sci. thesis was defended in 1966. A list of my works contains more than 90 titles.
Main publications:
Garkavi A. L. On the problem of best approximation of a function $f(x,y)$ by sums $\phi(ax+by)+\psi(cx+dy)$ // East J. Approximation, 1996, 2 (2), 151–154.
A. L. Garkavi, “On the approximation of functions in two variables by sums of the form $\varphi (x)+\psi (y)$ in the spaces $L_p$ ($1\le p\le \infty $)”, Sibirsk. Mat. Zh., 38:1 (1997), 56–68; Siberian Math. J., 38:1 (1997), 47–57
2.
A. L. Garkavi, “On Existence of Best Approximation of a Function of Two Variables by Sums of Plane Waves”, Trudy Mat. Inst. Steklova, 219 (1997), 130–136; Proc. Steklov Inst. Math., 219 (1997), 123–129
A. L. Garkavi, V. A. Medvedev, S. Ya. Havinson, “Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables”, Mat. Sb., 187:5 (1996), 3–14; Sb. Math., 187:5 (1996), 623–634
A. L. Garkavi, V. A. Medvedev, S. Ya. Havinson, “On existence of a best uniform approximation of a function in two variables by the sums $\varphi(x)+\psi(y)$”, Sibirsk. Mat. Zh., 36:4 (1995), 819–827; Siberian Math. J., 36:4 (1995), 707–713
A. L. Garkavi, “On minimization of integral functionals on the space of measurable functions”, Mat. Zametki, 54:5 (1993), 13–22; Math. Notes, 54:5 (1993), 1093–1099
1990
6.
A. L. Garkavi, V. A. Kaminskii, “Certain extremal problem in a measure space”, Mat. Zametki, 48:4 (1990), 19–28; Math. Notes, 48:4 (1990), 985–991
7.
A. L. Garkavi, “Algebraic and geometric components of Chebyshev rank of a system of vector-valued functions”, Sibirsk. Mat. Zh., 31:6 (1990), 39–45; Siberian Math. J., 31:6 (1990), 904–909
A. L. Garkavi, “Uniqueness of the best approximation in mean of vector-valued functions”, Mat. Zametki, 39:3 (1986), 337–348; Math. Notes, 39:3 (1986), 183–189
1985
10.
A. L. Garkavi, V. A. Shmatkov, “Uniqueness of a continuation of positive functionals”, Izv. Vyssh. Uchebn. Zaved. Mat., 1985, no. 9, 10–14; Soviet Math. (Iz. VUZ), 29:9 (1985), 10–15
1983
11.
A. L. Garkavi, “Estimating the norm of a conditional extension of a linear functional”, Mat. Zametki, 34:4 (1983), 559–572; Math. Notes, 34:4 (1983), 769–775
1981
12.
A. L. Garkavi, “Minimax balayage theorem and an inscribed ball problem”, Mat. Zametki, 30:1 (1981), 109–121; Math. Notes, 30:1 (1981), 542–548
1980
13.
A. L. Garkavi, “Method of cyclic descent in the problem of best approximation”, Mat. Zametki, 27:4 (1980), 549–558; Math. Notes, 27:4 (1980), 270–274
A. L. Garkavi, V. B. Gogolevskii, “On reliability of multicomputer systems”, Avtomat. i Telemekh., 1978, no. 4, 161–168
1975
15.
A. L. Garkavi, V. N. Zamyatin, “Conditional Chebyshev center of a bounded set of continuous functions”, Mat. Zametki, 18:1 (1975), 67–76; Math. Notes, 18:1 (1975), 622–627
A. L. Garkavi, V. A. Shmatkov, “On the Lamé point and its generalizations in a normed space”, Mat. Sb. (N.S.), 95(137):2(10) (1974), 272–293; Math. USSR-Sb., 24:2 (1974), 267–286
A. L. Garkavi, “The conditional Chebyshev center of a compact set of continuous functions”, Mat. Zametki, 14:4 (1973), 469–478; Math. Notes, 14:4 (1973), 827–831
A. L. Garkavi, “A certain criterion for a $K$-system of continuous functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1972, no. 4, 55–59
1971
19.
A. L. Garkavi, “The Helly problem and best approximation of summable functions”, Mat. Sb. (N.S.), 84(126):2 (1971), 196–217; Math. USSR-Sb., 13:2 (1971), 187–207
A. L. Garkavi, “The existence of a best approximating element in $(F)$-space with integral metric”, Mat. Zametki, 8:5 (1970), 583–594; Math. Notes, 8:5 (1970), 797–802
A. L. Garkavi, “Characteristic of Chebyshev subspaces of finite codimension in $L_1$”, Mat. Zametki, 7:2 (1970), 155–163; Math. Notes, 7:2 (1970), 95–99
A. L. Garkavi, “The theory of best approximation in normed linear spaces”, Itogi Nauki. Ser. Matematika. Mat. Anal. 1967, 1969, 75–132; Progr. Math., 8 (1970), 83–150
A. L. Garkavi, “The Helly problem and best approximation in a space of continuous functions”, Izv. Akad. Nauk SSSR Ser. Mat., 31:3 (1967), 641–656; Math. USSR-Izv., 1:3 (1967), 623–637
A. L. Garkavi, A. M. Flomin, “Approximative properties of sets in certain topological spaces”, Mat. Zametki, 2:5 (1967), 523–530; Math. Notes, 2:5 (1967), 809–812
25.
A. L. Garkavi, “The unit sphere of the space of polynomials with integral metric”, Mat. Zametki, 1:3 (1967), 299–304; Math. Notes, 1:3 (1967), 199–202
A. L. Garkavi, “Approximation properties of subspaces of finite defect in the space of continuous functions”, Dokl. Akad. Nauk SSSR, 155:3 (1964), 513–516
A. L. Garkavi, “A topology in Banach spaces”, Uspekhi Mat. Nauk, 18:1(109) (1963), 151–154
35.
A. L. Garkavi, “On best approximation by the elements of infinite-dimensional subspaces of a certain class”, Mat. Sb. (N.S.), 62(104):1 (1963), 104–120
A. L. Garkavi, “On the spaces of linear operators”, Dokl. Akad. Nauk SSSR, 132:3 (1960), 497–500
40.
A. L. Garkavi, “Simultaneous approximation to a periodic function and its derivatives by trigonometric polynomials”, Izv. Akad. Nauk SSSR Ser. Mat., 24:1 (1960), 103–128