For each $f\in L^p(ℝ)$ ($1\le p<\infty$) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$, a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^p(ℝ)$. There is an exchange theorem and inversion in norm.

For a Lebesgue integrable complex-valued function $f$ defined on ${ℝ}^{+}:=[0,\infty )$ let $\widehat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\widehat{f}\left(y\right)\to 0$ as $y\to \infty$. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1\left({ℝ}^{+}\right)$ there is a definite rate at which the Walsh-Fourier transform tends...

We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$-Hilbert series is a Vandermonde-like determinant. We show that the $h$-polynomial of the Grassmannian coincides with the $k$-Narayana polynomial. A simplified formula for the $h$-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$-Narayana numbers,...

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then...

Let $G$ be a compact group, let $n\in N\setminus \{0,1\}$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that ${\sigma }^n=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G\to C$ of the functional equation $$f\left(xy\right)+\sum _{k=1}^{n-1}f\left({\sigma }^k\left(y\right)x\right)=nf\left(x\right)f\left(y\right),x,y\in G,$$ in terms of unitary characters of $G$.

We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\widehat{f}\left(n\right)$, $n\in \mathbb{N}$, of functions $f\in L^p\left(G\right)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series ${\sum }_{n=1}^{\infty }{a}_n{\left|\widehat{f}\left(n\right)\right|}^r$, where $\left\{a_n\right\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative...

Some stronger and equivalent metrics are defined on $ℳ$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $ℳ$ are investigated.

Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $\xi$ of the sphere and we show that if $T$ has the value $\tau$ at $\xi$, then the Fourier-Laplace series of $T$ at $\xi$ is Abel-summable to $\tau$.

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator ${}_{a,b}$ by ${}_{a,b}\left(f\right)\left(z\right)=\Gamma (a+1)/\Gamma (b+1){\int }_0^1\left(f\left(t\right){(1-t)}^b\right)/\left({(1-tz)}^{a+1}\right)dt$, where a and b are non-negative real numbers. In particular, for a = b = β, ${}_{a,b}$ becomes the generalized Hilbert operator ${}_{\beta }$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that ${}_{a,b}$ is bounded on Dirichlet-type spaces $S^p$, 0 < p < 2, and on Bergman spaces $A^p$, 2 < p < ∞. Also we find an upper bound for the norm of the operator ${}_{a,b}$....

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱL^p{\left({ℝ}^d\right)}_{comp}$, 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱL^p{\left({ℝ}^d\right)}_{comp}$ if the mapping $x\mapsto {\nabla }_x\Phi (x,\eta )$ is constant on the fibres, of codimension r,...

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathscr{H}$, and $S$ be a selfadjoint operator in $\mathscr{H}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal ${𝒮}_p$ $(p>1),$ we derive a bound for ${\sum }_k{|\mathrm{R}{\lambda }_k\left(A\right)-{\lambda }_k\left(S\right)|}^p$, where ${\lambda }_k\left(A\right)$ $(k=1,2,\cdots )$ are the eigenvalues of $A$. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues...

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

Let $j\ge 2$ be a given integer. Let $H_k^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modulo group $\mathrm{SL}(2,\mathbb{Z})$. For $f\in H_k^{*}$, denote by ${\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s,{\mathrm{sym}}^{j}f)$) attached to $f$. We are interested in the average behaviour of the sum $$\sum _{\genfrac{}{}{0pt}{}{n=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x}{(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}}{\lambda }_{{\mathrm{sym}}^{j}f}^2\left(n\right),$$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

We first give a necessary and sufficient condition for $x^{-\gamma }\varphi \left(x\right)\in L^p$, 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either ${\sum }_{k=1}^{\infty }{a}_{k}coskx$ or ${\sum }_{k=1}^{\infty }{b}_{k}sinkx$, under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x)...

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n\in S_{k+1/2}(N,{\chi }_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$, where the Fourier coefficients $a\left(n\right)$ are real. Bruinier and Kohnen conjectured that the signs of $a\left(n\right)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${\left\{a\left(t{n}^2\right)\right\}}_n$, where $t$ is a squarefree integer such that $a\left(t\right)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that ${\left\{a\left(t{n}^2\right)\right\}}_n$ is equidistributed over any arithmetic progression $n\equiv dmodq$.

Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{\delta ,j}:\delta \in K̂₀,1\le j\le d_{\delta }$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_t$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{\delta }(i\lambda +\varrho )$ be the Kostant polynomials. We establish the following...