Fourier Series for Singular Measures
نویسندگان
چکیده
منابع مشابه
Fourier Series for Singular Measures
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [0, 1), every f ∈ L2(μ) possesses a Fourier series of the form f (x) = ∑n=0 cne. We show that the coefficients cn can be computed in terms of the quantities f̂ (n) = ∫ 1 0 f (x)e −2πinxdμ(x). We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ-bandlimited.
متن کاملFourier series for singular measures and the Kaczmarz algorithm
Using the Kaczmarz algorithm, we obtain a Fourier series formulation for functions in the L2 space of singular measures on the unit circle. This formula is applied to the problem of finding reproducing kernel Hilbert spaces inside the classical Hardy space, where the norm is instead that of boundary integration with respect to a singular measure. We also give some conditions ensuring that these...
متن کاملMoments and Legendre–Fourier Series for Measures Supported on Curves
Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures which is much easier to solve. However, an optimal solution μ of the latter solves the former if and only if the measure μ is supported on a “trajectory” {(t, x(t)) : t ∈ [0, T ]} for some measurable function x(t). We provide necessary and sufficient c...
متن کاملDetermination of a jump by Fourier and Fourier-Chebyshev series
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation. This is achieved without the finiteness assumption on the number of discontinuities. Two results on determination of ...
متن کاملFourier Series
for some fixed τ , which is called the period of f . Though function approximation using orthogonal polynomials is very convenient, there is only one kind of periodic polynomial, that is, a constant. So, polynomials are not good for approximating periodic functions. In this case, trigonometric functions are quite useful. A large class of important computational problems falls under the category...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Axioms
سال: 2017
ISSN: 2075-1680
DOI: 10.3390/axioms6020007