Two-sided Estimates Research Articles

Approximations With Polynomial, Trigonometric, Exponential Splines of the Third Order and Boundary Value Problem

This paper is devoted to the construction of localapproximations of functions of one and two variables using thepolynomial, the trigonometric, and the exponential splines. Thesesplines are useful for visualizing flows of graphic information.Here, we also discuss the parallelization of computations. Someattention is paid to obtaining two-sided estimates of theapproximations using interval analysis methods. Particularattention is paid to solving the boundary value problem by usingthe polynomial splines and the trigonometric splines of the thirdand fourth order approximation. Using the considered splines,formulas for a numerical differentiation are constructed. Theseformulas are used to construct computational schemes for solvinga parabolic problem. Questions of approximation and stability ofthe obtained schemes are considered. Numerical examples arepresented.

Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms

In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincar\'e inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the H\"older regularity of parabolic functions for symmetric non-local Dirichlet forms.

A Method for the Construction of LocalParabolic Splines with Additional Knots

We propose a general method for the construction of local parabolic splines with an arbitrary arrangement of knots for functions given on grid subsets of the number axis or its segment. Special cases of this scheme are Yu. N. Subbotin’s and B. I. Kvasov’s splines. For Kvasov’s splines, we consider boundary conditions different from those suggested by Kvasov. We study the approximating and smoothing properties of these splines in the case of uniform knots. In particular, we find two-sided estimates for the error of approximation of the function classes $$W_{\infty}^{2}$$ and $$W_{\infty}^{3}$$ by these splines in the uniform metric and calculate the exact uniform Lebesgue constants and the norms of the second derivatives on the class $$W_{\infty}^{2}$$ . These properties are compared with the corresponding properties of Subbotin’s splines.

Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ):E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.

Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

Using Duhamel’s formula, we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C1,1 domains. In addition, we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

Точные константы в двусторонней оценке С. А. Теляковского суммы ряда по синусам с выпуклой последовательностью коэффициентов

Известно, что сумма ряда по синусам $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$, коэффициенты которого образуют выпуклую последовательность $\mathbf b$, положительна на интервале $(0,\pi)$. Для оценки ее значений в окрестности нуля C. А. Теляковский использовал кусочно-непрерывную функцию $$ \sigma(\mathbf b,x)=\frac1{m(x)}\sum_{k=1}^{m(x)-1}k^2(b_k-b_{k+1}),\qquad m(x)=[\frac\pi x]. $$ Он показал, что разность $g(\mathbf b,x)-(b_{m(x)}/2)\operatorname{ctg}(x/2)$ в некоторой окрестности нуля допускает двустороннюю оценку через функцию $\sigma(\mathbf b,x)$ с абсолютными постоянными. В работе найдены точные значения этих постоянных на классе выпуклых последовательностей $\mathbf b$. Библиография: 8 названий.

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form H=−L+V; the free operator L is the generator of a symmetric Lévy process in Rd, d>1 (with non-degenerate jump measure) and V is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.

Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients

It is known that the sum of the sine series $$g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} $$bk sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function $$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].} $$ He showed that the difference g(b, x) − (bm(x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.

On Some Properties of Superreflexive Besov Spaces

This paper contains results concerning superreflective Besov spaces $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ . Namely, expressions for convexity moduli and smoothness moduli with respect to the “canonical” norms are derived, and properties related to the finite representability of Banach spaces and linear compact operators in $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ are examined. Additionally, inequalities of the Prus–Smarzewski type for arbitrary equivalent norms and inequalities of the James–Gurariy type are presented. Based on the latter, two-sided estimates for the norms of elements in $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ can be obtained in terms of the expansion coefficients of these elements in unconditional normalized Schauder bases.

Estimates of the Deviation from Exact Solutions of Boundary Value Problems in Measures Stronger than the Energy Norm

The paper is concerned with estimates of the difference between a given function and the exact solution of an elliptic boundary value problem. Estimates of this type have been derived earlier in terms of the natural energy norm. In this work, an approach is proposed to obtain stronger measures of the deviation and relevant estimates applicable if the exact solution and the approximation have additional regularity (with respect to the order of integrability). These measures include the standard energy norm as a simple special case. A general approach is proposed to construct various measures based on using an auxiliary variational problem. Two classes of measures whose properties are close to those of the $${{L}^{q}}$$ and $${{L}^{\infty }}$$ norms are studied in more detail. Their properties are established, and explicitly computable two-sided estimates (minorants and majorants) that involve only known functions are constructed.

Uniform pointwise asymptotics of solutions to quasi-geostrophic equation**The paper is partially supported by the NCN grant 2015/18/E/ST1/00239

We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in , α ∈ (1, 2). Furthermore, we give an upper bound of a similar type for any derivative of the solutions. Initial data in , p > 2/(α − 1), are also discussed.

Decay of Fourier Transforms and Generalized Besov Spaces

A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given.\ In particular, necessary and sufficient conditions of Titchmarsh type are obtained.\ The method is based on two-sided estimate for the rate of approximation of a $\beta$-admissible family of multipliers operators in terms of decay properties of Fourier transforms.

Cut ratios and Laplacian eigenvalues

We introduce local cut ratios as a sampling scheme for the cut ratio of a vertex subset in a graph. We prove a bound for the sampling error, as well as a two-sided estimate for the local cut ratio, in terms of Laplacian eigenvalues. We derive an inequality for the vertex cut ratio in terms of the (edge) cut ratio.

Estimates on the tail probabilities of subordinators and applications to general time fractional equations

In this paper, we study estimates on tail probabilities of several classes of subordinators under mild assumptions on the tails of their Lévy measures. As an application of that result, we obtain two-sided estimates for fundamental solutions of general homogeneous time fractional equations including those with Dirichlet boundary conditions.

Fractional Sturm–Liouville eigenvalue problems, I

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann–Liouville type. We then solve a Dirichlet type Sturm–Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann–Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann–Liouville integrals at those end-points. For each $$1/2<\alpha <1$$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $$\alpha \rightarrow 1^-$$, and that the fractional operator converges to an ordinary two term Sturm–Liouville operator as $$\alpha \rightarrow 1^-$$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $$\alpha $$.

Приближение функций частными суммами ряда Фурье по многочленам, ортогональным на произвольных сетках

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids $${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$$ of N points from segment [−1,1] with weight Δtj=tj+1−tj. Approximation properties of the constructed partial sums Sn,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function Ln,N(t) of considered discrete Fourier sums for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$$. The question of the convergence of Sn,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum Sn,N(f,t) from f(t) for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$$, which depends on n and the position of a point t ∊ [−1,1].

Second order monotone difference schemes with approximation on non-uniform grids for two-dimensional quasilinear parabolic convection-diffusion equations

The present communication is devoted to the construction of monotone difference schemes of the second order of local approximation on non-uniform grids in space for 2D quasi-linear parabolic convection-diffusion equation. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform norm C is proved. It is interesting to note that the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients.