On Determination of Best-Possible Constants in ... - Semantic Scholar
MATHEMATICS OCTOBER
OF COMPUTATION,
1980, PAGES
VOLUME
35, NUMBER
152
1191-1193
of Best-PossibleConstants On Determination in IntegralInequalitiesInvolvingDerivatives By Beny Neta Abstract. This paper is concernedwith the numericalapproximationof the best possible constantsay k in the inequality IIF(k)12 < y Yn,k{IIFI2 + lIF( )ll2}, where = 1JF112
foIF(x)12dx.
A list of all constants 'yn k for n < 10 is given.
1. Introduction. This paper utilizes the algorithmgivenin [1l to numerically approximatethe best possible constants'ynk' 1 < k < n, forn < 10 in the inequality:
1 IIF(k)112< Y2k {11F1 + IIFn) 112},
(1)
- 11denotes the L2 [0, o?) norm. The functionF has a locally absolutely conwhere 11 tinuous (n - l)st derivative. The inequality (1) is equivalent to IIF(k)II ? Mnlk IIFII(n-k)/nIIF(n)llkln,
(2)
where (3)
M2
Mn,k=en.k
kln
nk
k~~~
k
(n-k)ln
nk
see [1]. Interestin inequalities (1) and (2) increasedbecause of theirclose connection operator by bounded operwith problemsof best approximationof the differentiation ators; see [2], [3], [4], [5], and with the problem of best approximationof one class of functionsby another; see [4], [6], [7]. In the next section we shall give lower and upper bounds for the best possible constants-ynak and Mn,k for n < 10. 2. NumericalResults. In this section the best possible constants'Yn,k and Mnfk
are listed.
72i
Y31 = 732 =
see [1].
1,
=
3-2/
=
.555669,
Received October 31, 1978; revised February 1, 1980. Primary 46E30, 34B05, 65D20.
AMS (MOS) subject classifications (1970).
see [1] .
26A84; Secondary 47E05,
? 1980 American Mathematical Society 0025-571 8/80/0000-01 60/$01 .75 1191
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
1192
BENY NETA
Z8 - 6Z4 as the smallestpositivezero of the polynomial in [1], 741 is characterized Z4 2Z2 - 4Z + 1. 8Z2 + 1, and 742 is thesmallestpositivezero of thepolynomial UsingMuller'smethod[8] , we obtain741 = 743 = .339246, y42 = .225270. Remark. It is known,see [1], that
(4)
'yn,n-k=
forall n, k.
'Yn,k
tableof lowerand upperboundson in [1], one has the following Usingthe algorithm < < < < n 10 and 1 k [n/2]. For othervaluesof k, use (4). 'Yn,k for2 TABLE 1 IYn,k for 2 1
n\k 2
< n < 10, 1 < k < [n/2] 3
2
5
4
1.
3
.555669
4
.225271
.339246
5
(.225837,
.2258375)
(.102266,
.102268)
6
(.160328,
.160338)
(.051986,
.05199)
7
(.11936,
.11943)
(.028924,
.02895)
8
(.09128,
.09129)
(.0172,
.01723)
9
(.07593,
.07594)
(.010795,
.048)*
(.0068,
10
(.0479,
.0108) .007)
(.0361167,
.0361177) .0147)
(.014698, (.0068112,
.00681124)(.005014, .0036)
(.00345, (.0014163,
.0014165)
(.00193,
.0050145) .001938) .00068151)(.000642565,.00064257)
(.000681505,
tableof lower Using(3) and thevalueslistedin Table 1, one has the following and upperboundson Mnfk for2 < n < 10 and 1 < k < [n/2]. For othervaluesof k, useMnn-k = Mnfkforall n, k. 2
TABLE Mn k for2 < 1
n\k
n < 10, 1 < k < [n/2] 4
3
2
2
1.41421
3
2.07005
4
2.27432
9
(2.70248,
2.70249)
(4.37797,
4.37801)
6
(3.12838,
3.12848)
(6.02917,
6.02940)
(7.92662,
7.93019)
5
2.97963
(7.44141,
7.44151)
7
(3.55221,
3.55325)
8
(3.99579,
3.99601)
(10.09176,10.10056)
(16.86722,16.86727)
(19.97106,19.97206)
9
(4.32029,
4.32057)
(12.54043,12.54333)
(23.07295,23.23717)
(32.02543,32.09173)
10
(5.36995,
5.37555)
(15.35013,15.57423)
(36.06112,36.06367)
(53.62984,53.63004)
(11.60467,11.60546)
(55.78980,55.79001)
Remarks. 1. The lowerand upperboundsforeach n and k are givenin parenthesesand separatedby a comma,forexample,.11936 < Y7?1 < .11943. 2. The numberM4,2 in Table 2 agreeswiththatobtainedby Bradleyand Everitt[7] . 3. The numberM6,3 in thistableagreeswitha resultof Dawsonand Everitt[9]. of n. For fixedn the functions Conjecture.For fixedk the'Yn,k are decreasing of k up to k = [n/2]. functions 'Yn,k are decreasing
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
BEST POSSIBLE
CONSTANTS
IN INTEGRAL
INEQUALITIES
1193
Thustheinitialvalueof 'Ynk maybe takenin theinterval for n > 2
*,k = ? (0, yl-l,k)
by Kupcov,namely ratherthantheintervalsuggested In,k = (O, gn,k),
where gn, k =
n kk/n(n - k)(n-k)In
A. Zettland The authorwouldliketo thankProfessors Acknowledgements. himto the problem,and the refereeforhis suggestions M. K. Kwongforintroducing whichgreatlyimprovedthepaper. Department of Mathematical Sciences Northern Illinois University DeKalb, Illlinois 60115 1. N. P. KUPCOV, "Kolmogorov estimates for derivatives in L2(0, -o),"Proc. Steklov Inst. Math., v. 138, 1975, pp. 101-125. 2. S. B. STECKIN, "Inequalities between norms of derivatives of arbitrary functions," Acta Sci. Math. (Szeged), v. 26, 1965, pp. 225-230. 3. S. B. STECKIN, "Best approximation of linear operators," Mat. Zametki, v. 1, 1967, Math. Notes, v. 1, 1967, pp. 91-99. pp. 137-148 4. V. V. ARESTOV, "Sharp inequalities between the norms of functions and their derivatives," Acta Sci. Math. (Szeged), v. 33, 1972, pp. 243-267. 5. V. V. ARESTOV, "On some extremal problems for differentiable functions of one variable," Proc. Steklov Inst. Math., v. 138, 1975, pp. 1-29. 6. JU. N. SUBBOTIN & L. V. TAIKOV, "Best approximation of a differentiation operator in the space L2," Mat. Zametki, v. 3, 1968, pp. 157-164 = Math. Notes, v. 3, 1968, pp. 100-105. "On the inequality 1If 112< Kllf l - If 4)11," Quart. 7. J. S. BRADLEY & W. N. EVERITT, J. Math. Oxford Ser. (2), v. 25, 1974, pp. 241-252.
8. S. D. CONTE & C. DE BOOR, ElementaryNumericalAnalysis:An AlgorithmicApproach,
McGraw-Hill, New York, 1972.
9. E. R. DAWSON & W. N. EVERITT, On Recent Results in IntegralInequalities Involving Derivatives, Lecture given at the Ordinary and Partial Differential Equations Conference held at Dundee, Scotland, March 1976.
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
OF COMPUTATION,
1980, PAGES
VOLUME
35, NUMBER
152
1191-1193
of Best-PossibleConstants On Determination in IntegralInequalitiesInvolvingDerivatives By Beny Neta Abstract. This paper is concernedwith the numericalapproximationof the best possible constantsay k in the inequality IIF(k)12 < y Yn,k{IIFI2 + lIF( )ll2}, where = 1JF112
foIF(x)12dx.
A list of all constants 'yn k for n < 10 is given.
1. Introduction. This paper utilizes the algorithmgivenin [1l to numerically approximatethe best possible constants'ynk' 1 < k < n, forn < 10 in the inequality:
1 IIF(k)112< Y2k {11F1 + IIFn) 112},
(1)
- 11denotes the L2 [0, o?) norm. The functionF has a locally absolutely conwhere 11 tinuous (n - l)st derivative. The inequality (1) is equivalent to IIF(k)II ? Mnlk IIFII(n-k)/nIIF(n)llkln,
(2)
where (3)
M2
Mn,k=en.k
kln
nk
k~~~
k
(n-k)ln
nk
see [1]. Interestin inequalities (1) and (2) increasedbecause of theirclose connection operator by bounded operwith problemsof best approximationof the differentiation ators; see [2], [3], [4], [5], and with the problem of best approximationof one class of functionsby another; see [4], [6], [7]. In the next section we shall give lower and upper bounds for the best possible constants-ynak and Mn,k for n < 10. 2. NumericalResults. In this section the best possible constants'Yn,k and Mnfk
are listed.
72i
Y31 = 732 =
see [1].
1,
=
3-2/
=
.555669,
Received October 31, 1978; revised February 1, 1980. Primary 46E30, 34B05, 65D20.
AMS (MOS) subject classifications (1970).
see [1] .
26A84; Secondary 47E05,
? 1980 American Mathematical Society 0025-571 8/80/0000-01 60/$01 .75 1191
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
1192
BENY NETA
Z8 - 6Z4 as the smallestpositivezero of the polynomial in [1], 741 is characterized Z4 2Z2 - 4Z + 1. 8Z2 + 1, and 742 is thesmallestpositivezero of thepolynomial UsingMuller'smethod[8] , we obtain741 = 743 = .339246, y42 = .225270. Remark. It is known,see [1], that
(4)
'yn,n-k=
forall n, k.
'Yn,k
tableof lowerand upperboundson in [1], one has the following Usingthe algorithm < < < < n 10 and 1 k [n/2]. For othervaluesof k, use (4). 'Yn,k for2 TABLE 1 IYn,k for 2 1
n\k 2
< n < 10, 1 < k < [n/2] 3
2
5
4
1.
3
.555669
4
.225271
.339246
5
(.225837,
.2258375)
(.102266,
.102268)
6
(.160328,
.160338)
(.051986,
.05199)
7
(.11936,
.11943)
(.028924,
.02895)
8
(.09128,
.09129)
(.0172,
.01723)
9
(.07593,
.07594)
(.010795,
.048)*
(.0068,
10
(.0479,
.0108) .007)
(.0361167,
.0361177) .0147)
(.014698, (.0068112,
.00681124)(.005014, .0036)
(.00345, (.0014163,
.0014165)
(.00193,
.0050145) .001938) .00068151)(.000642565,.00064257)
(.000681505,
tableof lower Using(3) and thevalueslistedin Table 1, one has the following and upperboundson Mnfk for2 < n < 10 and 1 < k < [n/2]. For othervaluesof k, useMnn-k = Mnfkforall n, k. 2
TABLE Mn k for2 < 1
n\k
n < 10, 1 < k < [n/2] 4
3
2
2
1.41421
3
2.07005
4
2.27432
9
(2.70248,
2.70249)
(4.37797,
4.37801)
6
(3.12838,
3.12848)
(6.02917,
6.02940)
(7.92662,
7.93019)
5
2.97963
(7.44141,
7.44151)
7
(3.55221,
3.55325)
8
(3.99579,
3.99601)
(10.09176,10.10056)
(16.86722,16.86727)
(19.97106,19.97206)
9
(4.32029,
4.32057)
(12.54043,12.54333)
(23.07295,23.23717)
(32.02543,32.09173)
10
(5.36995,
5.37555)
(15.35013,15.57423)
(36.06112,36.06367)
(53.62984,53.63004)
(11.60467,11.60546)
(55.78980,55.79001)
Remarks. 1. The lowerand upperboundsforeach n and k are givenin parenthesesand separatedby a comma,forexample,.11936 < Y7?1 < .11943. 2. The numberM4,2 in Table 2 agreeswiththatobtainedby Bradleyand Everitt[7] . 3. The numberM6,3 in thistableagreeswitha resultof Dawsonand Everitt[9]. of n. For fixedn the functions Conjecture.For fixedk the'Yn,k are decreasing of k up to k = [n/2]. functions 'Yn,k are decreasing
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
BEST POSSIBLE
CONSTANTS
IN INTEGRAL
INEQUALITIES
1193
Thustheinitialvalueof 'Ynk maybe takenin theinterval for n > 2
*,k = ? (0, yl-l,k)
by Kupcov,namely ratherthantheintervalsuggested In,k = (O, gn,k),
where gn, k =
n kk/n(n - k)(n-k)In
A. Zettland The authorwouldliketo thankProfessors Acknowledgements. himto the problem,and the refereeforhis suggestions M. K. Kwongforintroducing whichgreatlyimprovedthepaper. Department of Mathematical Sciences Northern Illinois University DeKalb, Illlinois 60115 1. N. P. KUPCOV, "Kolmogorov estimates for derivatives in L2(0, -o),"Proc. Steklov Inst. Math., v. 138, 1975, pp. 101-125. 2. S. B. STECKIN, "Inequalities between norms of derivatives of arbitrary functions," Acta Sci. Math. (Szeged), v. 26, 1965, pp. 225-230. 3. S. B. STECKIN, "Best approximation of linear operators," Mat. Zametki, v. 1, 1967, Math. Notes, v. 1, 1967, pp. 91-99. pp. 137-148 4. V. V. ARESTOV, "Sharp inequalities between the norms of functions and their derivatives," Acta Sci. Math. (Szeged), v. 33, 1972, pp. 243-267. 5. V. V. ARESTOV, "On some extremal problems for differentiable functions of one variable," Proc. Steklov Inst. Math., v. 138, 1975, pp. 1-29. 6. JU. N. SUBBOTIN & L. V. TAIKOV, "Best approximation of a differentiation operator in the space L2," Mat. Zametki, v. 3, 1968, pp. 157-164 = Math. Notes, v. 3, 1968, pp. 100-105. "On the inequality 1If 112< Kllf l - If 4)11," Quart. 7. J. S. BRADLEY & W. N. EVERITT, J. Math. Oxford Ser. (2), v. 25, 1974, pp. 241-252.
8. S. D. CONTE & C. DE BOOR, ElementaryNumericalAnalysis:An AlgorithmicApproach,
McGraw-Hill, New York, 1972.
9. E. R. DAWSON & W. N. EVERITT, On Recent Results in IntegralInequalities Involving Derivatives, Lecture given at the Ordinary and Partial Differential Equations Conference held at Dundee, Scotland, March 1976.
This content downloaded from 205.155.65.226 on Wed, 14 Jan 2015 13:54:24 PM All use subject to JSTOR Terms and Conditions
