The inverse theorem in various metrics of approximation theory for periodic functions with monotone Fourier coefficients

N. A. Il'yasov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 153-162

Résumé

We prove the exactness with respect to order of an upper bound for the $k$th-order modulus of smoothness in $L_q({\mathbb T})$ in terms of the elements of a sequence of best approximations in $L_p({\mathbb T})$ on the class of all functions with monotonically decreasing Fourier coefficients, where $1$ and $k\in {\mathbb N}$.

Détail

Keywords: modulus of smoothness, best approximation, inverse theorem in various metrics, trigonometric Fourier series with monotone coefficients, order-sharp inequality on a class.

Bibliographie

[1] Ilyasov N.A., “O priblizhenii periodicheskikh funktsii srednimi Feiera–Zigmunda v raznykh metrikakh”, Mat. zametki, 48:4 (1990), 48–57 | MR | Zbl

[2] Ilyasov N.A., “Obratnaya teorema teorii priblizhenii v raznykh metrikakh”, Mat. zametki, 50:6 (1991), 57–65 | MR | Zbl

[3] Ilyasov N.A., “K obratnoi teoreme teorii priblizhenii periodicheskikh funktsii v raznykh metrikakh”, Mat. zametki, 52:2 (1992), 53–61 | MR | Zbl

[4] Geit V.E., “O tochnosti nekotorykh neravenstv v teorii priblizhenii”, Mat. zametki, 10:5 (1971), 571–582 | MR

[5] Geit V.E., “O strukturnykh i konstruktivnykh svoistvakh funktsii i ee sopryazhennoi v $L$”, Izv. vuzov. Matematika, 1972, no. 7(122), 19–30 | MR | Zbl

[6] Timan M.F., “Orthonormal systems satisfying an inequality of S.M. Nikol'ski”, Anal. Math., 4:1 (1978), 75–82 | DOI | MR | Zbl

[7] Ilyasov N.A., Teoremy vlozheniya dlya strukturnykh i konstruktivnykh kharakteristik funktsii, diss. ... kand. fiz.-mat. nauk, Baku, 1987, 150 pp.

[8] Simonov B., Tikhonov S., “Sharp Ul'yanov-type inequalities using fractional smoothness”, J. Approx. Theory, 162:9 (2010), 1654–1684 | DOI | MR | Zbl

[9] Ilyasov N.A., “Priblizhenie periodicheskikh funktsii srednimi Zigmunda”, Mat. zametki, 39:3 (1986), 367–382 | MR | Zbl

[10] Ilyasov N.A., “K pryamoi teoreme teorii priblizhenii periodicheskikh funktsii v raznykh metrikakh”, Tr. MIAN, 219 (1997), 220–234 | MR | Zbl

[11] Bari N.K., Trigonometricheskie ryady, Fizmatgiz, M., 1961, 936 pp. | MR

[12] Timan M.F., “O vlozhenii $L_p^{(k)}$ klassov funktsii”, Izv. vuzov. Matematika, 1974, no. 10(149), 61–74 | MR | Zbl

[13] Zigmund A., Trigonometricheskie ryady, v 2 t., v. 1, Mir, M., 1965, 615 pp. ; т. 2, 537 с. | MR

[14] Konyushkov A.A., “Nailuchshie priblizheniya trigonometricheskimi polinomami i koeffitsienty Fure”, Mat. sb., 44(86):1 (1958), 53–84

[15] Konyushkov A.A., “O nailuchshikh priblizheniyakh pri preobrazovanii koeffitsientov Fure metodom srednikh arifmeticheskikh i o ryadakh Fure s neotritsatelnymi koeffitsientami”, Sib. mat. zhurn., 3:1 (1962), 56–78 | Zbl

[16] Khardi G.G., Littlvud D.E., Polia G., Neravenstva, IL, M., 1948, 456 pp.

[17] Timan A.F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960, 624 pp.

[18] Timan M.F., “Obratnye teoremy konstruktivnoi teorii funktsii v prostranstvakh $L_p\ (1\le p\le \infty)$”, Mat. sb., 46(88):1 (1958), 125–132 | MR | Zbl

[19] Timan M.F., “O teoreme Dzheksona v prostranstvakh $L_p$”, Ukr. mat. zhurn., 18:1 (1966), 134–137 | MR | Zbl

[20] Kokilashvili V.M., “O priblizhenii periodicheskikh funktsii”, Tr. Tbilis. mat. in-ta, 34 (1968), 51–81 | Zbl

[21] Ulyanov P.L., “Teoremy vlozheniya i sootnosheniya mezhdu nailuchshimi priblizheniyami (modulyami nepreryvnosti) v raznykh metrikakh”, Mat. sb., 81(123):1 (1970), 104–131 | Zbl

[22] Edvards R., Ryady Fure v sovremennom izlozhenii, v 2 t., v. 1, Mir, M., 1985, 264 pp.; т. 2, 400 с.