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Journal articles on the topic 'Polynomials in Banach spaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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1

Gonz´lez, Manuel, and Joaquín M. Gutiérrez. "Unconditionally converging polynomials on Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 2 (March 1995): 321–31. http://dx.doi.org/10.1017/s030500410007314x.

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In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

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2

Kravtsiv, V. "Zeros of block-symmetric polynomials on Banach spaces." Matematychni Studii 53, no. 2 (June 24, 2020): 206–11. http://dx.doi.org/10.30970/ms.53.2.206-211.

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We investigate sets of zeros of block-symmetric polynomials on the direct sums of sequence spaces. Block-symmetric polynomials are more general objects than classical symmetric polynomials.An analogues of the Hilbert Nullstellensatz Theorem for block-symmetric polynomials on $\ell_p(\mathbb{C}^n)=\ell_p \oplus \ldots \oplus \ell_p$ and $\ell_1 \oplus \ell_{\infty}$ is proved. Also, we show that if a polynomial $P$ has a block-symmetric zero set then it must be block-symmetric.

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3

Peris, Alfredo. "Chaotic polynomials on Banach spaces." Journal of Mathematical Analysis and Applications 287, no. 2 (November 2003): 487–93. http://dx.doi.org/10.1016/s0022-247x(03)00547-x.

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4

Carando, Daniel. "Extendible Polynomials on Banach Spaces." Journal of Mathematical Analysis and Applications 233, no. 1 (May 1999): 359–72. http://dx.doi.org/10.1006/jmaa.1999.6319.

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5

FABIAN, M., D. PREISS, J. H. M. WHTTFIELD, and V. E. ZIZLER. "SEPARATING POLYNOMIALS ON BANACH SPACES." Quarterly Journal of Mathematics 40, no. 4 (1989): 409–22. http://dx.doi.org/10.1093/qmath/40.4.409.

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6

González, Manuel, and Joaquín M. Gutiérrez. "Orlicz-Pettis Polynomials on Banach Spaces." Monatshefte f�r Mathematik 129, no. 4 (May 9, 2000): 341–50. http://dx.doi.org/10.1007/s006050050080.

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7

Bu, Qingying, Gerard Buskes, and Yongjin Li. "Abstract M- and Abstract L-Spaces of Polynomials on Banach Lattices." Proceedings of the Edinburgh Mathematical Society 58, no. 3 (February 13, 2015): 617–29. http://dx.doi.org/10.1017/s0013091514000297.

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AbstractIn this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

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8

Yadav, Sarjoo Prasad. "On the denseness of Jacobi polynomials." International Journal of Mathematics and Mathematical Sciences 2004, no. 28 (2004): 1455–62. http://dx.doi.org/10.1155/s0161171204305314.

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LetXrepresent either a spaceC[−1,1]orLα,βp(w),1≤p<∞, of functions on[−1,1]. It is well known thatXare Banach spaces under the sup and thep-norms, respectively. We prove that there exist the best possible normalized Banach subspacesXα,βkofXsuch that the system of Jacobi polynomials is densely spread on these, or, in other words, eachf∈Xα,βkcan be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Explicit representation forf∈Xα,βkhas been given.

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9

Sarantopoulos, Yannis. "Bounds on the derivatives of polynomials on Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 2 (September 1991): 307–12. http://dx.doi.org/10.1017/s0305004100070389.

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AbstractWe generalize the classical Bernstein's and Markov's Inequalities for polynomials on any real Banach space. We also give estimates for the derivatives of homogeneous polynomials on real Banach spaces.

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10

Boyd, C., and R. A. Ryan. "THE NORM OF THE PRODUCT OF POLYNOMIALS IN INFINITE DIMENSIONS." Proceedings of the Edinburgh Mathematical Society 49, no. 1 (February 2006): 17–28. http://dx.doi.org/10.1017/s0013091504000756.

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AbstractGiven a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

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11

Anatriello, Giuseppina, Alberto Fiorenza, and Giovanni Vincenzi. "Banach function norms via Cauchy polynomials and applications." International Journal of Mathematics 26, no. 10 (September 2015): 1550083. http://dx.doi.org/10.1142/s0129167x15500834.

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Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial [Formula: see text]. We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.

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12

Brudnyi, Alexander. "Banach spaces of polynomials as “large” subspaces ofℓ∞-spaces." Journal of Functional Analysis 267, no. 4 (August 2014): 1285–90. http://dx.doi.org/10.1016/j.jfa.2014.05.006.

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13

Pellegrini, Leonardo. "Bases in spaces of homogeneous polynomials on Banach spaces." Journal of Mathematical Analysis and Applications 332, no. 1 (August 2007): 272–78. http://dx.doi.org/10.1016/j.jmaa.2006.09.058.

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14

H�jek, Petr. "Polynomials and injections of Banach spaces into superreflexive spaces." Archiv der Mathematik 63, no. 1 (July 1994): 39–44. http://dx.doi.org/10.1007/bf01196297.

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15

Lourenço, Mary L., and Luiza A. Moraes. "A class of polynomials from Banach spaces into Banach algebras." Publications of the Research Institute for Mathematical Sciences 37, no. 4 (2001): 521–29. http://dx.doi.org/10.2977/prims/1145477328.

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16

Hájek, Petr, and Michal Kraus. "Polynomials and identities on real Banach spaces." Journal of Mathematical Analysis and Applications 385, no. 2 (January 2012): 1015–26. http://dx.doi.org/10.1016/j.jmaa.2011.07.028.

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17

Kirwan, Pádraig, and Raymond A. Ryan. "Extendibility of homogeneous polynomials on Banach spaces." Proceedings of the American Mathematical Society 126, no. 4 (1998): 1023–29. http://dx.doi.org/10.1090/s0002-9939-98-04009-x.

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18

Defant, Andreas, and Pablo Sevilla-Peris. "Convergence of Dirichlet polynomials in Banach spaces." Transactions of the American Mathematical Society 363, no. 02 (February 1, 2011): 681. http://dx.doi.org/10.1090/s0002-9947-2010-05146-3.

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19

Velanga, T. "Ideals of polynomials between Banach spaces revisited." Linear and Multilinear Algebra 66, no. 11 (October 31, 2017): 2328–48. http://dx.doi.org/10.1080/03081087.2017.1394963.

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20

Llavona, Jose-G. "ZEROS OF REAL POLYNOMIALS ON BANACH SPACES." Journal of the Korean Mathematical Society 41, no. 1 (January 1, 2004): 77–94. http://dx.doi.org/10.4134/jkms.2004.41.1.077.

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21

Grecu, Bogdan C., and Raymond A. Ryan. "Polynomials on Banach spaces with unconditional bases." Proceedings of the American Mathematical Society 133, no. 4 (November 19, 2004): 1083–91. http://dx.doi.org/10.1090/s0002-9939-04-07738-x.

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22

Botelho, Geraldo, and Daniel M. Pellegrino. "Scalar-valued dominated polynomials on Banach spaces." Proceedings of the American Mathematical Society 134, no. 6 (December 20, 2005): 1743–51. http://dx.doi.org/10.1090/s0002-9939-05-08501-1.

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23

Aron, Richard M., and Petr Hájek. "Odd Degree Polynomials on Real Banach Spaces." Positivity 11, no. 1 (October 13, 2006): 143–53. http://dx.doi.org/10.1007/s11117-006-2035-9.

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24

MUÑOZ, GUSTAVO A., and YANNIS SARANTOPOULOS. "Bernstein and Markov-type inequalities for polynomials on real Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 3 (November 2002): 515–30. http://dx.doi.org/10.1017/s0305004102006217.

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In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space and provide estimates for the second and third derivatives of a polynomial on a real Banach space. Our result on a real Hilbert space answers a question raised by L. A. Harris in his commentary on problem 74 in the Scottish Book [20]. We also provide generalizations of previously obtained inequalities of the Bernstein and Markov-type for polynomials with curved majorants on a real Hilbert space.

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25

Agud, Lucia, Jose Manuel Calabuig, Maria Aranzazu Juan, and Enrique A. Sánchez Pérez. "Banach Lattice Structures and Concavifications in Banach Spaces." Mathematics 8, no. 1 (January 14, 2020): 127. http://dx.doi.org/10.3390/math8010127.

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Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

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26

Dineen, Seán. "Extreme integral polynomials on a complex Banach space." MATHEMATICA SCANDINAVICA 92, no. 1 (March 1, 2003): 129. http://dx.doi.org/10.7146/math.scand.a-14397.

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We obtain upper and lower set-theoretic inclusion estimates for the set of extreme points of the unit balls of $\mathcal{P}_{I}({}^{n}\!E)$ and $\mathcal{P}_{N}({}^{n}\!E)$, the spaces of $n$-homogeneous integral and nuclear polynomials, respectively, on a complex Banach space $E$. For certain collections of Banach spaces we fully characterise these extreme points. Our results show a difference between the real and complex space cases.

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27

Verkalets, N. B., and A. V. Zagorodnyuk. "On geometric extension of polynomials on Banach spaces." Carpathian Mathematical Publications 5, no. 2 (December 30, 2013): 196–98. http://dx.doi.org/10.15330/cmp.5.2.196-198.

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28

Choi, Yun Sung, Domingo Garcia, Sung Guen Kim, and Manuel Maestre. "THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE." Proceedings of the Edinburgh Mathematical Society 49, no. 1 (February 2006): 39–52. http://dx.doi.org/10.1017/s0013091502000810.

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AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

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29

Lacruz, Miguel. "Norms of polynomials and capacities on Banach spaces." MATHEMATICA SCANDINAVICA 85, no. 2 (December 1, 1999): 271. http://dx.doi.org/10.7146/math.scand.a-18276.

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30

Kavadjiklis, Andreas, and Sung Guen Kim. "Plank type problems for polynomials on Banach spaces." Journal of Mathematical Analysis and Applications 396, no. 2 (December 2012): 528–35. http://dx.doi.org/10.1016/j.jmaa.2012.06.029.

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31

Fernández-Unzueta, Maite. "Dunford-Pettis and Dieudonné polynomials on Banach spaces." Illinois Journal of Mathematics 45, no. 1 (January 2001): 291–307. http://dx.doi.org/10.1215/ijm/1258138269.

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32

Cilia, Raffaella, and Joaquín M. Gutiérrez. "Integral polynomials on Banach spaces not containing ℓ1." Czechoslovak Mathematical Journal 60, no. 1 (February 23, 2010): 221–31. http://dx.doi.org/10.1007/s10587-010-0011-9.

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33

Lacruz, M., and A. M. Tonge. "Polynomials on Banach Spaces: Zeros and Maximal Points." Journal of Mathematical Analysis and Applications 192, no. 2 (June 1995): 539–42. http://dx.doi.org/10.1006/jmaa.1995.1188.

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34

Ferrer, Jesús. "On Banach Spaces Whose Unit Sphere Determines Polynomials." Acta Mathematica Sinica, English Series 23, no. 1 (September 12, 2006): 175–88. http://dx.doi.org/10.1007/s10114-005-0832-x.

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35

Cilia, Raffaella, and Joaquín M. Gutiérrez. "Nonexistence of certain universal polynomials between Banach spaces." Journal of Mathematical Analysis and Applications 427, no. 2 (July 2015): 962–76. http://dx.doi.org/10.1016/j.jmaa.2015.02.072.

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36

Lacruz, Miguel. "Norms of polynomials and capacities on Banach spaces." Integral Equations and Operator Theory 34, no. 4 (December 1999): 494–99. http://dx.doi.org/10.1007/bf01272887.

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37

Verkalets, N., and A. Zagorodnyuk. "Linear Subspaces in Zeros of Polynomials on Banach Spaces." Journal of Vasyl Stefanyk Precarpathian National University 2, no. 4 (December 24, 2015): 105–36. http://dx.doi.org/10.15330/jpnu.2.4.105-136.

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38

Aguiar, Adriano L., and Luiza A. Moraes. "Reflexivity of Spaces of Polynomials on Direct Sums of Banach Spaces." Publications of the Research Institute for Mathematical Sciences 45, no. 2 (2009): 351–61. http://dx.doi.org/10.2977/prims/1241553122.

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39

Zagorodnyuk, Andriy, and Anna Hihliuk. "Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability." Axioms 10, no. 3 (July 7, 2021): 150. http://dx.doi.org/10.3390/axioms10030150.

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In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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40

Cilia, Raffaella, Maria D'Anna, and Joaquín M. Gutiérrez. "Polynomials on banach spaces whose duals are isomorphic to ℓ1 (Γ)." Bulletin of the Australian Mathematical Society 70, no. 1 (August 2004): 117–24. http://dx.doi.org/10.1017/s0004972700035863.

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We prove that the dual of a Banach space E is isomorphic to an ℓ1(Γ) space if and only if, for a fixed integer m, every m-homogeneous 1-dominated polynomial on E is nuclear. This extends a result for linear operators due to Lewis and Stegall. The same techniques used for this result allow us to prove that, if every m-homogeneous integral polynomial between two Banach spaces is nuclear, then every integral (linear) operator between the same spaces is nuclear.

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41

Valdivia, Manuel. "Banach Spaces of Polynomials without Copies of l 1." Proceedings of the American Mathematical Society 123, no. 10 (October 1995): 3143. http://dx.doi.org/10.2307/2160673.

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42

Aron, R., C. Boyd, and Y. S. Choi. "Unique Hahn-Banach theorems for spaces of homogeneous polynomials." Journal of the Australian Mathematical Society 70, no. 3 (June 2001): 387–400. http://dx.doi.org/10.1017/s1446788700002408.

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AbstractWe investigate certain norm and continuity conditions that provide us with ‘uniqe Hahn-Banch Theorems’ from (nc0) to (nℓ∞) and from N(nE) to N(nE″). We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on complex c0 to ℓ∈ but there is no unique norm-preserving extension from (3c0) to (3ℓ∈).

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43

Dineen, Seán, and Jorge Mujica. "Banach spaces of homogeneous polynomials without the approximation property." Czechoslovak Mathematical Journal 65, no. 2 (June 2015): 367–74. http://dx.doi.org/10.1007/s10587-015-0181-6.

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44

Pinasco, Damián, and Ignacio Zalduendo. "Tchakaloff’s theorem and $K$-integral polynomials in Banach spaces." Proceedings of the American Mathematical Society 145, no. 8 (January 25, 2017): 3395–408. http://dx.doi.org/10.1090/proc/13520.

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45

Verkalets, Nadiia. "Maximal zero-subspaces of diagonal polynomials on Banach spaces." International Journal of Mathematical Analysis 10 (2016): 711–18. http://dx.doi.org/10.12988/ijma.2016.6465.

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46

Floret, Klaus. "Minimal ideals of n-homogeneous polynomials on Banach spaces." Results in Mathematics 39, no. 3-4 (May 2001): 201–17. http://dx.doi.org/10.1007/bf03322686.

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47

Botelho, Geraldo, and Daniel Pellegrino. "Absolutely summing polynomials on Banach spaces with unconditional basis." Journal of Mathematical Analysis and Applications 321, no. 1 (September 2006): 50–58. http://dx.doi.org/10.1016/j.jmaa.2005.08.017.

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48

Muñoz, Gustavo, Yannis Sarantopoulos, and Andrew Tonge. "Complexifications of real Banach spaces, polynomials and multilinear maps." Studia Mathematica 134, no. 1 (1999): 1–33. http://dx.doi.org/10.4064/sm-134-1-1-33.

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49

Pryimak, H. M. "Homomorphisms and functional calculus in algebras of entire functions on Banach spaces." Carpathian Mathematical Publications 7, no. 1 (July 3, 2015): 108–13. http://dx.doi.org/10.15330/cmp.7.1.108-113.

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In the paper the homomorphisms of algebras of entire functions on Banach spaces to a commutative Banach algebra are studied. In particular, it is proposed a method of constructing of homomorphisms vanishing on homogeneous polynomials of degree less or equal than a fixed number $n$.

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50

Bombal, Fernando, and Ignacio Villanueva. "Regular multilinear operators on C(K) spaces." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 11–20. http://dx.doi.org/10.1017/s0004972700033281.

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The purpose of this paper is to characterise the class of regular continuous multilinear operators on a product of C(K) spaces, with values in an arbitrary Banach space. This class has been considered recently by several authors in connection with problems of factorisation of polynomials and holomorphic mappings. We also obtain several characterisations of a compact dispersed space K in terms of polynomials and multilinear forms defined on C(K).

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