Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Suppose that d ≥ 1 is an integer, ${\alpha \in (0,d)}$ is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1) ^d . The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.     

Quasiuniversal Fourier–Walsh Series for the Classes Lp[0, 1], p > 1

Mathematical Notes - It is proved that, for each number p > 1, there exists a function L1[0, 1] whose Fourier–Walsh series is quasiuniversal with respect to subseries-signs in the...     

Upper Bounds of the Lebesgue Constants for Fourier–Jacobi Series Summation Methods Defined by a Multiplier Function

Let be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators generated by a multiplier matrix defined by the following relations:

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...     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ 10.     

Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

In 1987 Harris proved (Proc Am Math Soc 101(4):637–643, 1987)—among others—that for each \(1\le p<2\) there exists a two-dimensional function \(f\in L^p\) such that its triangular Walsh–Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or (C, 1)) means of the triangle two variable Walsh–Fourier series of \(L^1\) functions. Namely, we prove the a.e. convergence \(\sigma _n^{\bigtriangleup }f = \frac{1}{n}\sum _{k=0}^{n-1}S_{k, n-k}f\rightarrow f\) (\(n\rightarrow \infty \)) for each integrable two-variable function f.     

(C,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces H _p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H _p to L _p (1/( + 1) < p < ) and is of weak type (L ₁,L ₁). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L ₁ converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H _p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H ₁ and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H ₁ ,L ₁). Consequence, the Walsh—Fourier series of every function f H ₁ is (C,,) summable to the function f.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

Strong Summability of Two-Dimensional Vilenkin–Fourier Series

Ukrainian Mathematical Journal - We study the exponential uniform strong summability of two-dimensional Vilenkin–Fourier series. In particular, it is proved that the two-dimensional...     

Representation of Functions in the Weighted Space L 1 μ by Trigonometric and Walsh Series

We prove that there exists a series of the form (*)

The Fourier–Faber–Schauder Series Unconditionally Divergent in Measure

We prove that, for every ε ∈ (0, 1), there is a measurable set E ? [0, 1] whose measure |E| satisfies the estimate |E| > 1?ε and, for every function f ∈ C_[0,1], there is ? f ∈ C_[0,1] coinciding with f on E whose expansion in the Faber–Schauder system diverges in measure after a rearrangement.     

Upper Bounds for Neumann–Schatten Norms

Let H and H _aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,,s>0 and J a bounded linear transformation from the Hilbert space D(H ^s/2) (equipped with the graph scalar product of H ^s/2) to H _aux. It is shown that the operator J(H+)^–t belongs to the Neumann–Schatten class of order p=2+2(u–t)/(t–s/2) provided s/2<t<u,t–s/2<u–t and J(H+)^–u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+)^–t is derived. If J is a closed operator from D(H ^1/2) to H _aux and D(J)D(H) then there exists a unique self-adjoint operator H ^J in H such that D(H ^J )D(J) and ( . Conditions which are sufficient in order that the operator (H ^J +)^–1–(H+)^–1 is compact and conditions which are sufficient in order that the wave operators W ^±(H ^J ,H) exist and are complete are derived. Instead of (Jf,Jg)_aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (–) ^r in L ²(R ^d ) for any strictly positive real number and H ^J equals (–) ^r + for some suitably chosen measure is discussed in detail. In particular, new results on existence and completeness of the wave operators W ^±(–+,–) are obtained.     

Strong approximation by Marcinkiewicz means of two-dimensional Walsh–Kaczmarz–Fourier series

In this paper we study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series. In particular, it is proved that the Marcinkiewicz type of two-dimensional Walsh–Kaczmarz–Fourier series of every continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is the best possible.     

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).     

Fourier–Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds

We develop and implement a semi-numerical method for computing high-order Taylor approximations of unstable manifolds for hyperbolic fixed points of compact infinite-dimensional maps. The method can follow folds in the embedding and describes precisely the dynamics on the manifold. In order to ensure the accuracy of our computations in spite of the many truncation and round-off errors, we develop a posteriori error bounds for the approximations. Deliberate control of round-off errors (using interval arithmetic) in conjunction with explicit analytical estimates leads to mathematically rigorous computer-assisted theorems describing precisely the truncation errors for our approximation of the invariant manifold. The method is applied to the Kot-Schaffer model of population dynamics with spatial dispersion.