Induced inner product structures and Cauchy-Schwarz inequalities for linear functionals
Email:
trangnh@utc.edu.vn
Từ khóa:
Linear functionals, inner product, moment matrix, moment sequence, Cauchy-Schwarz inequality.
Tóm tắt
Linear functionals on finite-dimensional polynomial spaces generate fundamental algebraic and analytic structures, including moment sequences, moment matrices, and functional inequalities. Associated with a linear functional on polynomials of bounded degree is a moment matrix whose entries are given by the values of the functional on products of monomials and naturally exhibit a Hankel structure. Adopting an intrinsic functional-analytic viewpoint, this paper studies linear functionals on polynomial spaces without invoking any external representation framework. We develop a unified algebraic setting in which linearity, positivity-type conditions, moment matrices, and a functional inequality are examined simultaneously. We distinguish properties arising purely from linearity from those requiring additional structural assumptions. In particular, we establish a Cauchy-type inequality for linear functionals under mild algebraic conditions, independent of any a priori inner product structure. Under stronger positivity assumptions, the linear functional induces an inner product on polynomial spaces, with the associated moment matrix reflecting this structure precisely. Moreover, the Hankel structure of moment matrices is clarified as an intrinsic consequence of polynomial multiplication.Tài liệu tham khảo
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[2]. J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990.
[3]. J. A. Shohat, J. D. Tamarkin, The Problem of Moments, American Mathematical Society, Providence, RI, 1943.
[4]. N. I. Akhiezer, The Classical Moment Problem, Oliver & Boyd, Edinburgh, 1965.
[5]. R. E. Curto, L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston Journal of Mathematics, 17 (1991) 603–635.
[6]. R. E. Curto, L. A. Fialkow, Flat Extensions of Positive Moment Matrices: Relations in Analytic or Conjugate Terms, in: J. B. Conway, B. B. Morrel (eds.), Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, 1st ed., Springer, Basel, (1998) 59–82. https://doi.org/10.1007/978-3-0348-8779-3.
[7]. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978.
[8]. M. Laurent, Sums of Squares, Moment Matrices and Optimization over Polynomials, in: M. Putinar, S. Sullivant (eds.), Emerging Applications of Algebraic Geometry, Springer, New York, (2008) 157–270. https://doi.org/10.1007/978-0-387-09686-5.
[9]. J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, Singapore, 2010.
[10]. S. Fitzpatrick, Linear Algebra: A Second Course, Featuring Proofs and Python, University of Lethbridge, Lethbridge, 2023.
[11]. K. Schmüdgen, The Moment Problem, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-64546-9.
[12]. G. Strang, Linear Algebra and Its Applications, 4th ed., Thomson Brooks/Cole, Belmont, 2006.
[13]. P. Karageorgis, Chapter 3. Bilinear forms. https://www.maths.tcd.ie/~pete/ma1212/chapter3.pdf, 2024, (accessed April 13, 2026).
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Nhận bài
28/01/2026
Nhận bài sửa
14/04/2026
Chấp nhận đăng
22/04/2026
Xuất bản
15/05/2026
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Kiểu trích dẫn
Nguyen Ha, T. (1778778000). Induced inner product structures and Cauchy-Schwarz inequalities for linear functionals. Tạp Chí Khoa Học Giao Thông Vận Tải, 77(4), 530-541. https://doi.org/10.47869/tcsj.77.4.14
