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This volume contains the proceedings of the AMS Special Sessions on Frames, Wavelets and Gabor Systems and Frames, Harmonic Analysis, and Operator Theory, held from April 16-17, 2016, at North Dakota State University in Fargo, North Dakota. The papers appearing in this volume cover frame theory and applications in three specific contexts: frame constructions and applications, Fourier and harmonic analysis, and wavelet theory.

The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces.

This volume contains the proceedings of the AMS Special Sessions on Frames, Wavelets and Gabor Systems and Frames, Harmonic Analysis, and Operator Theory, held from April 16-17, 2016, at North Dakota State University in Fargo, North Dakota.The papers appearing in this volume cover frame theory and applications in three specific contexts: frame constructions and applications, Fourier and harmonic analysis, and wavelet theory.

In this paper, we introduce and study a new concept of orthogonality, namely, mean p-angular distance orthogonality in normed linear spaces. We investigate main properties of this type of orthogonality and its relation to some other previously defined generalized orthogonalities. Then, westudy α-existence, α-diagonal existence and S-existence theorems for this type of orthogonality. Moreover, a new characterization of inner product spaces is given in terms of property (H), properly formulated for the mean p-angular distance orthogonality.

Given 𝑝, 𝑞 ∈ ℝ, we generalize the classical Dunkl—Williams inequality for 𝑝-angular and 𝑞-angular distances in inner product spaces. We extend the Hile inequality for arbitrary 𝑝-angular and 𝑞-angular distances and study some geometric aspects of a generalization of Dunkl—Williams inequality. Introducing power refinements, we show significant power refinements of the generalized Dunkl—Williams inequality under some mild conditions. Among other things, we give new characterizations of inner product spaces with regard to 𝑝-angular and 𝑞-angular distances. In particular, we prove that if 𝑝,𝑞,𝑟 ∈ ℝ, 𝑞 ≠ 0 and 0 ≤ 𝑝/𝑞 < 1, then 𝑋 is an inner product space if and only if for every 𝑥,𝑦 ∈ 𝑋 \\ {0}, ‖ ‖ x ‖ p − 1 x − ‖ y ‖ p − 1 y ‖ ≤ 2 1 / r ‖ ‖ x ‖ q − 1 x − ‖ y ‖ q − 1 y ‖ [ ‖ x ‖ r ( q − p ) + ‖ y ‖ r ( q − p ) ] 1 / r .

Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable.This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.

Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces X satisfying the following property: there is a function \\varepsilon\\to \\Delta_X(\\varepsilon) tending to zero with \\varepsilon0 such that every operator T\\colon \\ L_2\\to L_2 with \\|T\\|\\le \\varepsilon that is simultaneously contractive (i.e., of norm \\le 1) on L_1 and on L_\\infty must be of norm \\le \\Delta_X(\\varepsilon) on L_2(X). The author shows that \\Delta_X(\\varepsilon) \\in O(\\varepsilon^\\alpha) for some \\alpha0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \\theta-Hilbertian spaces for some \\theta0 (see Corollary 6.7), where \\theta-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).

The notion of the sub-exact sequence is the generalization of exact sequence in algebra especially on a module. A module over a ring R is a generalization of the notion of vector space over a field F. Refers to a special vector space over field F when we have a complete inner product space, it is called a Hilbert space. A space is complete if every Cauchy sequence converges. Now, we introduce the sub-exact sequence on Hilbert space which can later be useful in statistics. This paper aims to investigate the properties of the sub-exact sequence and their relation to direct summand on Hilbert space. As the result, we get two properties of isometric isomorphism sub-exact sequence on Hilbert space.

This study defines the concept of soft 2–normed space. The concepts of Cauchy sequence and convergent sequence in soft 2–normed spaces have been considered. It is demonstrated that every convergent sequence is a Cauchy sequence in 2–normed spaces. Furthermore, it is demonstrated that a convergent sequence possesses a unique limit. Additionally, the concept of soft 2-inner product space is introduced and examined its important properties. This is followed by the demonstration of the Cauchy-Schwarz inequality and the Parallelogram law within these spaces and the convergence of sequences in a soft 2– inner product space is analyzed. Finally, the definition of the soft 2-bilinear functional is provided, along with the definitions of orthogonality and b-best approximation, which are derived from this definition.