바티의 L1-수렴성 연구에 관한 소고

On Weighted Integrability of Trigonometric Series and L 1 -Convergence of Fourier Series

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is Lp-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In the present article , We have extended the result of S. Tikhonov[8] and used y-general monotone coefficients to prove the L^1 -convergence of the trigonometric series. Also,"||g-Sn||≡o(1)" considered as , the necessary and sufficient condition for the convergence in terms of coefficients.

Chapter 12 - Fourier Series and Integrals

Many results introduced for the absolutely convergence of Fourier series in terms of absolutely continuous functions. Here we study the convergence of Fourier series in terms of $p$-integrable functions series.

We obtain sharp conditions for the absolute uniform convergence of Fourier series in the eigenfunctions of the Schrodinger operator with Kato potential in a bounded domain for functions lying in the domains of generalized fractional powers of the original Schrodinger operator or in generalized Besov classes with a sharp exponent.

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

In this paper, we show a complete characterization of the uniform boundedness of the partial sum operator in a discrete Sobolev space with Jacobi measure. As a consequence, we obtain the convergence of the Fourier series. Moreover it is showed that this Sobolev space is the first category which implies that it is not possible to apply the Banach–Steinhaus theorem.

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We writebeing the complex Fourier series of f (θ).

Introduction to Circuit Analysis and Design

Trajectory equations are considered for classical three-particle problems on a straight line with potentials that are homogeneous coordinate functions. It is proposed to consider the case of a zero total energy as a completely integrable one in a generalized sense, since in it the order of the differential trajectory equation is lowered to the first one, the variables in the Hamiltonian-Jacobi equation are separated, there are additional first integral and invariant tori, in spite of the fact that the system cannot be integrable by the Liouville-Arnold theorem. The solutions are constructed (i) in a parametric form, (ii) in the form of a convergent perturbative series of a new type, and (iii) as a convergent Fourier series.

In this paper, we consider numerical and trigonometric series with a very general monotonicity condition. First, a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting. In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum.

Further Topics on the Convergence of Fourier Series

For a function f~L~ we denote the n-th partial sum of the Fourier series with respect to the system ~(~'~) by S(~'~)(]; x) . See [i, 2] for the properties of Fourier series with respect to orthonormal systems. Le t us s e t I ( x ) ~ x and m0 = {0, t/2, t/4, 3/4, t/8,...}. This c l a s s o f sy s t ems was i n t r o d u c e d by F r a n k l i n i n 1928. He has shown t h a t t he s y s tem ~(~,x) i s a b a s i s i n C(X) ( s e e [ 1 ] ) . Sys tems ~(~.~) f o r ~, d i f f e r e n t f rom I , were f i r s t c o n s i d e r e d by Reed [ 3 ] . He has c o n s t r u c t e d an i n f i n i t e l y d i f f e r e n t i a b l e ~ and a d i f f e r e n t i a b l e f , f o r wh ich S ~ ~ d i v e r g e s a t a p o i n t . I n t h i s n o t e we w i l l o b t a i n some r e s u l t s on t h e p rob l em o f c o n v e r g e n c e o f S~"~)(f ; x) a l m o s t e v e r y w h e r e and in t he m e t r i c o f L~. Namely, we will prove the following theorem. THEOREM i. If ~ is absolutely continuous and a function /~L ~ , then S~~ x) converges to f(x) almost everywhere On X with respect to the measure V. Moreover, we can explicitly indicate a set of full measure, on which the convergence holds trivially. More precisely, the following theorem is valid. THEOREM i' If an approximately continuous nonzero ~'(x) exists at a point xo~X and moreover xo is a Lebesgue point for f in L~, then S~=~)(/; x) converges at xo to f(xo). Here, by a Lebesgue point for a function f in L~ we mean a point xo satisfying the condition