Heat Properties for Groups

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)

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.

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.

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).

For a separable finite diffuse measure space {mathcal {M}} and an orthonormal basis {varphi _n} of L^2({mathcal {M}}) consisting of bounded functions varphi _nin L^infty ({mathcal {M}}), we find a measurable subset Esubset {mathcal {M}} of arbitrarily small complement |{mathcal {M}}{setminus } E|<epsilon , such that every measurable function fin L^1({mathcal {M}}) has an approximant gin L^1({mathcal {M}}) with g=f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of {mathcal {M}}=G/H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

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.

Further Topics on the Convergence of Fourier Series

The <TEX>$L^1$</TEX>-convergence of Fourier series problems through additional assumptions for Fourier coefficients were presented by W. H. Young in 1913. We say that they are the classical results. Using modified trigonometric series is the convenience method to study the <TEX>$L^1$</TEX>-convergence of Fourier series problems. they are called the neoclassical results. This study concerns with the <TEX>$L^1$</TEX>-convergence of Fourier series. We introduce the classical and neoclassical results of <TEX>$L^1$</TEX>-convergence sequentially. In particular, we investigate <TEX>$L^1$</TEX>-convergence results focused on the results of Bhatia's studies. In conclusion, we present the research minor lineage of Bhatia's studies and compare the classes of <TEX>$L^1$</TEX>-convergence mutually.

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

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n n -dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log ⁡ L L\log L . In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.