Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ? l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).     

On the L1-convergence and behavior of coefficients of Fourier–Vilenkin series

In this paper, we prove the following statement that is true for both bounded and some type of unbounded Vilenkin systems: for any \( \varepsilon \in (0,1)\), there exists a measurable set \(E\subset [0,1)\) of measure bigger than \(1-\varepsilon \) such that for any function \(f \in L^{1}[0,1)\), it is possible to find a function \(g\in L^{1}[0,1)\) coinciding with f on E, Fourier series of g with respect to Vilenkin system are convergent in \(L^{1}\)-norm and the absolute values of non zero Fourier coefficients of g are monotonically decreasing.     

On the Fourier coefficients of Siegel–Eisenstein series of degree 2 for odd level

The Ramanujan Journal - We calculate the Fourier coefficients of the Siegel–Eisenstein series of degree 2, level $$p^n$$ with trivial character. The main result is to compute the Siegel...     

On Changes of Variable that Preserve the Absolute Convergence of Fourier–Haar Series of Continuous Functions

Mathematical Notes - In this paper, we discuss the problem of the number of representations of positive integers as sums of triangular numbers. The method we use is similar to Rankin’s way in...     

On the dimension of Chowla–Milnor space

In a recent work, Gun, Murty and Rath defined the Chowla?CMilnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of ??(k)/ ?? ^k for odd positive integers k?>?1. In this paper, we give an alternate proof of their theorem about the conditional lower bound.     

On the dimension of the space of ℝ-places of certain rational function fields

We prove that for every n ∈ ? the space M(K(x ₁, …, x _n ) of ?-places of the field K(x ₁, …, x _n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x ₁, …, x _n )) ≤ n. For n = 2 the space M(K(x ₁, x ₂)) has covering and integral dimensions dimM(K(x ₁, x ₂)) = dim_? M(K(x ₁, x ₂)) = 2 and the cohomological dimension dim _G M(K(x ₁, x ₂)) = 1 for any Abelian 2-divisible coefficient group G.     

On the zeros of the Davenport–Heilbronn function

Let N ₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N ₀(T) ? T (ln T)^1/2+1/16?ε, where ε is an arbitrarily small positive number.     

On the cohomology of the Losev–Manin moduli space

We determine the cohomology of the Losev–Manin moduli space ${\overline{M}_{0, 2 | n}}$ of pointed genus zero curves as a representation of the product of symmetric groups ${\mathbb{S}_2 \times \mathbb{S}_n}$ .     

On the mean convergence of Fourier–Jacobi series

The convergence of Fourier–Jacobi series in the spaces L _p,A,B is studied in the case where the Lebesgue constants are unbounded.     

On absolute convergence of Haar—Fourier series of superpositions of functions

В работе решены некот орые задачи, поставле нные П. Л. Ульяновым [5]. ПустьA _χ обозначает мн ожество тех функций?∈L(0, l), ряды Фурье которых вс юду абсолютно сходятся. Если? — вещественная функция на интервале [a, b], то для того, чтобы для каж дой?∈A _χ удовлетворяющей ус ловиюa≦?(t)≦b, 0≦t≦1, выполнялось включен ие?○?∈A _χ , необходимо и достато чно, чтобы? была лине йной функцией, т.е.?(x)=cx+d,x∈[a,b], гдеc иd — вещественны е постоянные. То же самое утвержден ие остается в силе, есл и заменить классA _? наA _χ ?C[0, 1]. Пусть далее? непреры вно дифференцируема я функция на [a,b]. Если некоторый мо дуль непрерывностиω и мод уль непрерывностиω(δ;?′) удовлетворяют услов ию \(\sum\limits_{n = 0}^\infty {\omega (\omega (2^{ - n} } ;\varphi '))\omega (2^{ - n} )< \infty ,\) , то?○?∈A _χ для каждой функци и?∈A _χ ∩H ^ω , гдеa≦?(t)≦b, 0≦t≦1.     

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Mathematical Notes - Let $$f(x)$$ be a function belonging to the Lebesgue class $$L^p({\mathbb R}_+)$$ on the semiaxis $${\mathbb R}_+=[0,+\infty)$$ , $$1\le p\le 2$$ , and let $$\widehat{f}$$ be...     

Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

The Ramanujan Journal - In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the...     

On the coefficients of the asymptotic expansion of the kernel of Berezin–Toeplitz quantization

We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin?CToeplitz quantization obtained recently by Ma?CMarinescu, and of the composition of two Berezin?CToeplitz quantizations. Our main tool is the stationary phase formula of Melin?CSj?strand.     

On the divergence of double Fourier–Walsh and Fourier–Walsh–Kaczmarz series of continuous functions

Acta Mathematica Hungarica - We prove that there exists a continuous function on $$[0,1]^2$$ , with a certain smoothness, whose double Fourier–Walsh series diverges on a set of positive...     

On the first simultaneous sign change for Fourier coefficients associated with Hecke–Maass forms

The Ramanujan Journal - Let f and g be two distinct Hecke–Maass cusp forms of weight zero for $$SL(2,\mathbb {Z})$$ with Laplacian eigenvalues $$\frac{1}{4}+u^{2}$$ and $$\frac{1}{4}+v^{2}$$...     

Fourier coefficients of functions from many-dimensional classes of bounded Λ-variation

Estimates of trigonometric Fourier coefficients are obtained for functions from the classes of bounded many-dimensional Λ-variation and functions continuous in Λ-variation. In some cases it is proved that these estimates are unimprovable by the order.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.