In this note, we investigate orthogonal expansions adapted to exponential weights and function spaces endowed with Hermite-frequency decompositions. Our objective is to formulate the Hausdorff–Young inequality within the framework of Hermite based Triebel–Lizorkin type spaces. The results rest on two underlying principles. First, since the Hermite functions {ℋ_m(x)} , along with their generalized counterparts, are particular instances of generalized Freud functions associated with generalized Freud weights, we rely on the L^∞ and L^q , q<∞ , estimates of Kasuga and Sakai which extend Hille’s estimate |ℋ_m(x)|≲ m^-1/12 , m=1,2,… Second, we apply interpolation techniques involving the Orlicz–Lorentz spaces Λ (φ _X, C) , where C is a Young (or concave) function and φ _X denotes the fundamental function of a rearrangement-invariant space X. This approach enables us to derive Hausdorff–Young inequalities in L^p , Lorentz, Orlicz, and Λ (φ _X, C) based Hermite Triebel–Lizorkin spaces. Finally, in the Coda we present a general Hausdorff–Young inequality for the generalized Freud coefficients of functions f∈ L^p_0(ℝ)+L^p_1(ℝ) , where 1≤ p_0

We compute the integral of monomials of the form x2β over the unit sphere and the unit ball in Rn where β = (β1, . . . , βn) is a multi–index with real components βk > −1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, βk → ∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in Rn. We also consider the Fourier transform of monomials xα restricted to the unit sphere in Rn, where the multi–indices α have integer components, and discuss their behaviour at the origin.

We establish maximal inequalities for integral operators on ℝ^n with bounded kernel which allow for the pointwise evaluation of these operators, including the Fourier transform, for functions in Lorentz and Orlicz spaces. We introduce and estimate the maximal Fourier coefficients with respect to a uniformly bounded ONS on L^2_μ (I) for functions in L^p_μ (I) , with 1< p≤ 2 , where μ is a positive regular measure on a closed bounded interval I in the line. We obtain an improved Hausdorff–Young inequality with the maximal Fourier coefficients in place of the Fourier coefficients for 1

We compute the integral of monomials of the form x^2β over the unit sphere and the unit ball in R^n where β = (β_1,...,β_n) is a multi-index with real components β_k > -1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, β_k →∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in R^n. We also consider the Fourier transform of monomials x^α restricted to the unit sphere in R^n, where the multi-indices α have integer components, and discuss their behaviour at the origin.

We establish maximal inequalities for integral operators on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}<^>n$$\end{document} with bounded kernel which allow for the pointwise evaluation of these operators, including the Fourier transform, for functions in Lorentz and Orlicz spaces. We introduce and estimate the maximal Fourier coefficients with respect to a uniformly bounded ONS on L mu 2(I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2_\mu (I)$$\end{document} for functions in L mu p(I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p_\mu (I)$$\end{document}, with 1 ((u+1)(m-1)del u-u(u+1)(q-1)del v)in Omega x(0,T), v(t)=Delta v-v+ui n Omega x(0,T)(EOT*) in a ball Omega subseteq mathbb R <^> 2 with n >= 2/2 Previous results show that unbounded solutions exist for all m. q m - q < (z - 2)/n which, however, are necessarily global in time if q <= 0 It is expected that finite-up is possible whenever q > 6 but in the fully parabolic setting this has so far only been shown when max(m, ql >= 1 the present paper, we substantially extend these findings. Our main results for the two-and three-dimensional settings state that (*) admits solutions blowing up in finite time if that is, also for certain m, q with max[m, ql < 1 As a key new ingredient in our proof, we make use of (singular) pointwise upper m-q 2/3 m>2/3if n =2, n=3. that is, also for certainm,qwith max{m,q}<1. As a key new ingredient in our proof, wemake use of (singular) pointwise upper estimates foru

We discuss the Hausdorff–Young inequality in the context of maximal integral estimates, including the case of Hermite and Laguerre expansions. We establish a maximal inequality for integral operators with bounded kernel on ℝ , which in particular allows for the pointwise evaluation of these operators, including the Fourier transform, for functions in appropriate Lorentz and Orlicz spaces. In the case of the Hermite expansions we prove a refined Hausdorff–Young inequality, further sharpened by considering the maximal Hermite coefficients in place of the Hermite coefficients when estimating the appropriate Lorentz and Orlicz norms. We also consider the refined companion Hausdorff–Young inequality and Hardy–Littlewood type inequalities for the Hermite expansions. Similar results are proved for the Laguerre expansions.

We discuss a sharpened Hausdorff–Young inequality and estimate the maximal coefficients of orthogonal expansions in terms of Freud polynomials when 1

In Chap. 2 we introduce the Π–Riemann integral and discuss its basic properties. In particular, we observe that even though the Riemann sums of an integrable function converge to the integral, they may do so arbitrarily slowly. It is often the case that we take the limit of sums closely related to the Riemann sums of f, and we are thus led to Bliss’ theorem. We also discuss various numerical methods, or quadrature rules, to evaluate definite integrals, with special emphasis in the rate of approximation. The methods include the right and left Riemann sums, the uneven tag sums, and the midpoint, trapezoid, and Simpson rules. These methods also allow for the computation of roots of nonlinear equations. And we observe that if the function f in question is convex, or concave, the upper and lower sums converge to the integral monotonically.

This book uncovers the full capabilities of the Riemann integral by applying a modern view. Discover new results and historical insights.

We discuss a sharpened Hausdorff-Young inequality for n-dimensional Hermite expansions.

In Chap. 4 we introduce the modified Π–Riemann sums. Some of the applications discussed include ψ–asymptotically distributed sequences, uniformly distributed sequences, and extensions of recent results on deleting items and disturbing mesh in the Riemann integral.

We describe the Lorentz space $L(p, r), 0 < r < p, p > 1$, in terms of Orlicz type classes of functions L . As a consequence of this result it follows that Stein's characterization of the real functions on $R^n$ that are differentiable at almost all the points in $R^n$, is equivalent to the earlier characterization of those functions given by A. P. Calderon.

We consider general formulations of the change of variable formula for the Riemann-Stieltjes integral, including the case when the substitution is not invertible.