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Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that satisfies mild conditions, and emerges as a generalization of the Gram-Schmidt process. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to the introduction of the Wronski basis.

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the discrete Fourier transform, and we identify general conditions for it to be invertible when applied to any real or complex sequence. Applications to the mathematical sciences are also presented.

Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the definition and properties of a new space of functions, which will be called multivariate Taylor measure function space. The proposed generalization emerges as a unifying framework that includes many concepts from mathematics as special cases.

We present methods that provide all zeroes and extrema of a function that do not require differentiation. Using point process theory, we are able to describe the locations of zeroes or maxima, their number, as well as their distribution over a given window of observation. The algorithms in order to accomplish the theoretical development are also provided, and they are exemplified using many illustrative examples, for real and complex functions.

Based on the Wronski determinant, we propose the construction of linearly independent orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of the functions under consideration, that satisfies minimal properties. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to what we call the Wronski basis.

We propose and study a novel collection of signed measures, which will be apply called Taylor measures. Stochastic versions of the new measures are also defined and studied. We illustrate, through examples, how the deterministic and stochastic versions of the proposed Taylor measures emerge as a unifying framework that includes many concepts from mathematics and probability theory as special cases.