Local Extremums of Trigonometric Polynomial

Local Extremums of Trigonometric Polynomial

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Datum:2007
1. Verfasser: Belov, I.S.
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Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/7609
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Zitieren:Local extremums of trigonometric polynomial / I.S. Belov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 291-297. — Бібліогр.: 1 назв. — англ.

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spelling irk-123456789-76092010-04-07T12:00:38Z Local Extremums of Trigonometric Polynomial Belov, I.S. 2007 Article Local extremums of trigonometric polynomial / I.S. Belov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 291-297. — Бібліогр.: 1 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/7609 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Belov, I.S.
spellingShingle Belov, I.S.
Local Extremums of Trigonometric Polynomial
author_facet Belov, I.S.
author_sort Belov, I.S.
title Local Extremums of Trigonometric Polynomial
title_short Local Extremums of Trigonometric Polynomial
title_full Local Extremums of Trigonometric Polynomial
title_fullStr Local Extremums of Trigonometric Polynomial
title_full_unstemmed Local Extremums of Trigonometric Polynomial
title_sort local extremums of trigonometric polynomial
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2007
url http://dspace.nbuv.gov.ua/handle/123456789/7609
citation_txt Local extremums of trigonometric polynomial / I.S. Belov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 3. — С. 291-297. — Бібліогр.: 1 назв. — англ.
work_keys_str_mv AT belovis localextremumsoftrigonometricpolynomial
first_indexed 2025-07-02T10:25:29Z
last_indexed 2025-07-02T10:25:29Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2007, vol. 3, No. 3, pp. 291�297 Local Extremums of Trigonometric Polynomial I.S. Belov National Technical University "KhPI" 21 Frunze Str., Kharkiv, 61002, Ukraine E-mail:bigor@kpi.kharkov.ua Received March 23, 2005 Local extremes of the trigonometric polynomial Bn(�) = k=2nX k=n sin k� k are considered, and various inequalities between them are proved. In par- ticular, the greatest and the least values of Bn(�) are found. Key words: trigonometric polynomial, local extreme, averaging, arrange- ment of zeroes, numerical analysis, logarithmic derivative. Mathematics Subject Classi�cation 2000: 26A09. Let us consider a trigonometric polynomial Tmn(�) = sinm� m + : : :+ sinn� n with a derivative T 0 mn(�) = cosm� + : : : + cosn�: The local extremes Tmn(�), obviously, are among the points �p : sin n�m+ 1 2 �p = 0 and �q : cos n+m 2 �q = 0: For m = 1, the sequences f�pg and f�qg alternate, thus in points �p they are local minima, and in points �q they are local maxima. Moreover, local maxima decrease in q, and for An(�) = T1;n(�) we have ([1, p. 91]) An(�) � An( � n+ 1 ); 0 � � � �: (1) In the paper the similar results for local extreme of a polynomial Bn(�) = Tn;2n(�) are obtained. This question arises, for example, when considering the series 1X n=0 �nT2n;2n+1�1; �n = �1: c I.S. Belov, 2007 I.S. Belov In the proof of estimation (1) in ([1, p. 293]) the following method of averaging is used. Let a; b be two local extremes of An(�). Consider An(b)�An(a) = bZ a A0 n(�)d� = � c = a+ b 2 ; d = b� a 2 � = cZ a [A0 n(�) +A0 n(� + d)]d�: If the sum [A0 n(�)+A0 n(�+d)] preserves a sign in the interval [a; c], we shall obtain the sign of di�erence An(b)�An(a). Let us put Bn(�) = sinn� n + : : :+ sin 2n� 2n ; B0 n(�) = cos 3n 2 � sin n+ 1 2 � sin �=2 : (2) As the cos 3n�=2 frequency in (2) is approximately three times more than the sin(n+1)�=2 one, it is convenient to consider the cos 3n�=2 zeroes in the interval (0; �) by groups of three, and we denote aq = � 3n + 2� n (q � 1); bq = aq + 2� 3n = � n + 2� n (q � 1); cq = bq + 2� 3n = � � n + 2� n q; �q = 2� n+ 1 q; q � series; q = 1; : : : ; n+ 1 2 : The last series is incomplete if n is odd. In this case b[(n+1)=2] = �[(n+1)=2] = �. It is easy to see that the relative arrangement of zeroes in a q-series is as follows: aq < bq < cq < �q; � 1 � q � n+ 1 6 � ; seriesA; aq < bq < �q < cq; � n+ 1 6 < q � n+ 1 2 � ; seriesB: Theorem 1. Bn(aq) > Bn(bq), q = 1; : : : ; (n+ 1)=2: P r o o f. In this case the averaging is not necessary, because in the interval (aq � � � bq) the functions cos 3n�=2 and sin(n+ 1)�=2 preserve the sign and cos 3n 2 � = (�1)qÆ1; sin n+ 1 2 � = (�1)q�1Æ2; Æi > 0: Let �q; �q+1; �q+2; �q+3 be the four consecutive zeroes of cos 3n�=2 and sn(�) = B0 n(�) +B0 n � � + 2� 3n � +B0 n � � + 4� 3n � ; �q � � � �q+1; be the averaging of the derivative B0 n(�) in the interval (�q; �q+3). Using obvious statements 292 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 Local Extremums of Trigonometric Polynomial Lemma 1. sin � � + 2� 3 � + sin � = sin � � + � 3 � and Lemma 2. sin� sin(� + h)� sin(�+ h) sin� = sinh sin(�� �); it is easy to obtain an explicit expression for sn(�): Lemma 3. sn(�) = sin � 3n cos 3n 2 � sin � 2 sin � � 2 + � 3n � sin � � 2 + 2� 3n � � � p 3 cos � n� 2 + � 3 � sin � 2 +2 sin � 3n sin n� 2 cos � � 2 + � 3n �� = s1 + s2: (3) The unobtrusive advantage of representation (3) in comparison with (2) is in a regular position of zeroes. The cos � � 2 + � 3n � zeroes are in points aq and the sinn�=2 zeroes are in points (cq + aq+1)=2. It follows, in particular, that in the intervals (aq; bq) and (bq; cq) these functions preserve the sign. At the same time, with the growth of q, the sin (n+ 1)� 2 zeroes move from the interval (cq; aq+1) to the interval (bq; cq). Theorem 2. Bn(aq) > Bn(aq+1), q = 1; : : : ; [(n� 1)=2]. P r o o f. By Lemma 3, it is enough to check the inequality sn(�) = s1 + s2 � 0; aq � � � bq: (4) Let us prove that both terms in (4) are nonpositive. It follows from the relations cos 3n� 2 = (�1)qÆ1; sin n� 2 = (�1)q�1Æ2; cos � n� 2 + � 3 � = (�1)qÆ3; Æi > 0; that can easily be checked. Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 293 I.S. Belov Theorem 3. Bn(bq) < Bn(bq+1); q = 1; : : : ; [(n� 1)=2]: P r o o f. In view of Lem. 3, it is enough to check the inequality sn(�) = s1 + s2 � 0; bq � � � cq: (5) A nonnegativity of both summands in (5) follows from the relations cos 3n� 2 = (�1)q�1Æ1; sin n� 2 = (�1)q�1Æ2; cos � n� 2 + � 3 � = (�1)qÆ3; Æi > 0; since cos 3n�=2 changes the sign, but sin(n�)=2 and cos � n� 2 + � 3 � preserve it. Theorem 4. Bn(cq) > Bn(cq+1); q = 1; : : : ; [n�1 2 ]. P r o o f. As in the proof of Th. 2, it is enough to check the inequality sn(�) = s1 + s2 � 0; aq � � � cq+1: (6) Similarly to the case above, it follows from the relations cos 3n� 2 = (�1)q�1Æ1; cos( n� 2 + � 3 ) = (�1)qÆ2; Æi > 0; that s1 � 0. At the same time, sin n� 2 as well as s2 in (6) changes the sign in the point (cq + aq+1)=2 = q2�=n. Moreover, the numerical analysis shows that for small q the sum s1 + s2 near the point aq+1 takes positive values. Therefore, for the estimation of Z aq+1 cq s2(�)d� one more averaging is necessary. For h 2 (0; � 3n ) we put �1 = 2�=q�h; �2 = 2�=q+h. Then cos 3n�1=2 = cos 3n�2=2. Therefore, for �(h) = s2(�1) + s2(�2) we have �(h) = s2(�1) + s2(�2) = 2 sin2 � 3n cos 3n 2 �1 sin n 2 �1 � 2 4 cos � �1 2 + � 3n � sin �1 2 sin � �1 2 + � 3n � sin � �1 2 + 2� 3n � + cos � �2 2 + � 3n � sin �2 2 sin � �2 2 + � 3n � sin � �2 2 + 2� 3n � 3 5 < 0; 294 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 Local Extremums of Trigonometric Polynomial since �1 < �2. Finally, aq+1Z cq s2(�)d� = cq+aq+1 2Z cq s2(�)d� + aq+1Z cq+aq+1 2 s2(�)d� = 2� n qZ 0 �(h)dh < 0; and Theorem 4 is proved. Theorem 5. Bn(Aq) > Bn(cq); q = 1; : : : ; [(n+ 1)=6]. P r o o f. In this case, for aq � � � bq we put sn(�) = B0 n(�) +B0 n(� + 2�=3n) = cos 3n 2 � sin � 2 sin( � 2 + � 3n ) � 2 sin n+ 1 2 � sin � 6n cos � � 2 + � 6n � �2 sin �� 6 + � 6n � sin � 2 cos � (n+ 1) � 2 + � 6 + � 6n �� : (7) As cos 3n�=2 = (�1)qÆ, Æ > 0, aq � � � bq; it is enough to check the positivity of the square brackets, multiplied by (�1)q�1, that is equivalent to the inequality sin n+ 1 2 � � sin � 6n cos � � 2 + � 6n � + 2 sin2 �� 6 + � 6n � sin � 2 � � sin �� 3 + � 3n � sin � 2 cos(n+ 1) � 2 ; (8) with odd q, and we have the opposite inequality for even q. For de�niteness we consider the case with an odd q. In (8) let us omit a positive term 2 sin2 �� 6 + � 6n � � sin � 2 . Then we are restricted with the interval aq � � � � n+ 1 + 2� n (q � 1), because cos(n+ 1)�=2 � 0 holds on the interval � n+1 + 2� n (q � 1) � �, and (8) is obvious. After division by positive sin(n+ 1)�=2, in the case of odd q we obtain the inequality sin � 6n cos � � 2 + � 6n � sin � 2 � sin �� 3 + � 3n � cot (n+ 1)� 2 ; or sin � 3n cot � 2 � sin �� 3 + � 3n � cot (n+ 1)� 2 � 2 sin2 � 6n : (9) Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 295 I.S. Belov Lemma 4. cot (n+1)� 2 cot � 2 decreases monotonously when aq � � � � n+ 1 + 2� n (q � 1): P r o o f. A nonnegativity of the logarithmic derivative of fraction follows immediately from the well-known inequality j sin(n�)j � nj sin �j and from the positivity sin(n+ 1)� when aq � � � � n+ 1 + 2� n (q � 1). By Lemma 4, cot (n+ 1)� 2 � cot(n+ 1)�=6n cot �=6n cot �=2 and inequality (9) follows from the inequality cot � 2(n+ 1) [1 + cos � 3n � sin �� 3 + � 3n � cot(n+ 1) � 6n ] � sin � 3n : The last inequality, except, perhaps, some initial values n, is given by the following Lemma 5. sin �� 3 + � 3n � cot(n+ 1) � 6n � 3 2 : The proof of Lem. 5 follows from the inequality sin �� 3 + 2� � cot �� 6 + � � � 3=2; 0 � � � � 2 ; with � = �=6n. The case of even q is considered in a similar way, and Th. 5 is completely proved. The relation between Bn(bq) and Bn(cq), q = 1; : : : ; [(n + 1)=6], is a little more complicated. Using the averaging over zero �q = 2� n+ 1 q, similar to that applied in Th. 4, we receive Proposition 1. If �q = 2�q=(n + 1) > (bq + cq)=2, that holds when 0 � q � (n� 1)=3, then Bn(bq) < Bn(cq), otherwise Bn(cq) < Bn(bq). From Theorems 3 and 4 it follows that Bn(bq) monotonously increases in q, and Bn(cq) monotonously decreases. The following theorem is valid. Theorem 6. Bn(b0) < Bn(c[n�1 2 ]). P r o o f. Really, Bn(b0) = Bn �� n � = 2nX k=n sin k� n k = [k = l + n] = � nX l=0 sin l� n l + n ' � Z 1 0 sin(�t) l + t dt = �0:433785: 296 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 Local Extremums of Trigonometric Polynomial On the other hand, for example, when n is even Bn � c j n�1 2 j � = Bn � � � � 3n � = 2nX k=n sin(k� � k� 3n ) k = 2nX k=n (�1)k+1 sin k� 3n k = (�1)n+1 nX l=0 (�1)l sin(� 3 + l� 3n ) l + n = (�1)n+1 n 2X l=0 " sin(� 3 + l� 3n ) l + n � sin(� 3 + (l+1)� 3n ) l + 1 + n # : As ����� " sin(� 3 + l� 3n ) l + n � sin(� 3 + (l+1)� 3n ) l + 1 + n #����� � 2 sin � 3n 1 + n + 1 (l + n)(l + n+ 1) ; then ���Bn � c[n�1 2 ] ���� � 2 ln 2 sin � 3n + 1 2n ; Bn(b0) < Bn � c[n�1 2 ] � ; n > 6: For initial values n the inequality is checked by direct calculation. Theorem 6 is proved. As above, let �q = 2�q=(n+ 1), q = 1; : : : ; [(n� 1)=2], be the zeroes of sin(n+ 1)�=2 in the q-series. The following assertion is valid. Proposition 2. Bn(�q) � Bn(cq); Bn(�q) � Bn(aq+1); Bn(cq) � Bn(aq+1); 1 � q � [(n+ 1)=6] ; Bn(cq) � Bn(�q); Bn(cq) � Bn(aq+1); Bn(�q) � Bn(aq+1); [(n+ 1)=6] < q � [(n� 1)=2] : (10) The �rst and the second inequalities in (10) can be easily checked without averaging as in Th. 1. The last inequalities can be proved similarly to Th. 5, but the proofs are more cumbersome because of the di�erence of denominators in �q and cq, aq+1. References [1] G. Polia and G. Szego, Tasks and Theorems from Analysis. 2. Gostechteoretizdat, Moscow, 1956. Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 3 297

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