Moduli of continuity and average decay of Fourier transforms: two-sided estimates
Dimitri Gioev
备注：
15 pages, 1 figure, case of dim=1 added, to appear in: Integable systems, random matrices, and applications: Conference in honor of Percy Deift's 60th birthday (New York, NY, 2006), Contemp. Math., Amer. Math. Soc
We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that our approach does not use a regularly varying comparison function as in [2]. A corollary of Theorem 1.1 deals with the equivalence of the two-sided estimates on the modulus of continuity on one hand, and on the tail of the Fourier transform, on the other (Corollary 1.5). This corollary is applied in the proof of the violation of the so-called entropic area law for a critical system of free fermions in [4,5].
We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.
We study the moduli of continuity of functions of bounded variation and of their variation functions. It is easy to see that the modulus of continuity of a function of bounded variation is always smaller or equal to the modulus of continuity of its variation function. We show that we cannot make any reasonable conclusion on the modulus of continuity of the variation function if we only know the modulus of continuity of the parent function itself. In particular, given two moduli of continuity, the first being weaker than Lipschitz continuity, we show that there exists a function of bounded vari…
We obtain asymptotic estimates for the best approximations by trigonometric polynomials in the metric space $C$ $(L_p)$ of classes of periodic functions that can be represented as a convolution of kernels $Ψ_β$, which Fourier coefficients tend to zero faster than any power sequence, with functions $φ\in C (φ\in L_p),$ which moduli of continuity do not exceed a fixed majorant $ω(t)$. It is proved that in the spaces $C$ and $L_1$ the obtained estimates are asymptotically exact for convex moduli of continuity $ω(t)$.
Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an $L^2$ function, in the setting of Riemannian symmetric spaces of rank $1$ whose growth depends on a $k$th-order modulus of continuity.
This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n >= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion…
We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain $Ω\subset\mathbb{R}^N$. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: "Those domains, whose unit normal is well defined and has Lipschitz constant one." We also offer an extension to infinitely dimensional spaces $L^p$, $p\in (1,\infty)$.
We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is …
Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono, we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where T'(x) is infinite have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of T are also obtained.
In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. In particular, we improve estimates of Kato \cite{Ka} and show that $$ \big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\df(S^*S)^{1/2}$. Moreover, we show that this inequality is sharp. We prove in this paper that if $f$ is a nondecreasing continuous function on $\R$ that vanishes on $(…
The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity of functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The author builds continuous linear mappings of such spaces of functions defined on cube $ I^d, $ to ordinary Nikolskii and Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are function extension operators, thus incurring coincidence of both kinds of spaces on the cube $ I^d. $
We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_α( Mf) \le (1 + α)^{-1}\operatorname{Lip}_α( f)$, $α\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic beha…
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundar…
We consider a sequence of composite Bernstein operators and the quadrature formulae associated with them. Upper bounds for the approximation error of continuous functions and for the approximation of integrals of continuous functions are given. The bounds are described in terms of moduli of continuity of order one and two. Two inequalities of Tchebycheff-Grüss-type are also included.
Let $Ω\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (i.e., quantitatively open and path-connected) satisfiying the capacity density condition. Let $L_0 u=-\mathrm{div}(A_0 \nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real uniformly elliptic operators in $Ω$, with $ω_{L_0}, ω_L$ the associated elliptic measures. We establish the equivalence between the following properties: (i) $ω_L \in A_{\infty}(ω_{L_0})$, (ii) $L$ is $L^p(ω_{L_0})$-solvable for some $p\in (1,\infty)$, (iii) bounded null solutions of $L$ satisfy Carleson measure estimates with respec…
We formulate some special conditions for the integrable functions and moduli of continuity. We give the results on rate of approximation of such functions by matrix means of their Fourier series, where the entries of the rows of the matrix generate the sequences belonging to the classes MRBVS and MHBVS. We also present some results on norm approximation for functions from the generalized integral Lipschitz classes.
We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selectio…
