Journal of Approximation Theory最新文献

We study the circumradius of a random section of an ${\ell }_p$-ellipsoid, $0<p\le \infty$, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if $1\le p\le \infty$ and compressed sensing if $0<p\le 1$. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an ${\ell }_p$-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and $1\le p\le \infty$, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value $1-\frac{1}{p}$ or not. If $1\le p\le 2$, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021] for the case $p=2$.

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling’s approximation $n!=\sqrt{2\pi }{n}^{n+1/2}{e}^{-n}{e}^{r_n}.$ This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.

We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone $V^{d+1}$ and its surface $V_0^{d+1}$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator $\Delta$. Concretely, we treat the generalized Charlier weights in the framework of $\Delta$-Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role.

In this paper, we give a strong uniqueness characterization theorem for the Chebyshev center of a set of infinitely many functions relative to a finite-dimensional linear space on a compact Hausdorff space. Additionally, we derive an alternation theorem for Chebyshev centers relative to a weak Chebyshev space on any compact set of the real line. Furthermore, we show an intrinsic characterization of those linear spaces where an alternation theorem holds.

In this paper we discuss a conjecture of Schumaker that the principal submatrices of collocation matrices of bivariate Bernstein polynomials over triangular grids have positive determinant. It is easy to show that the conjecture holds for the 2 × 2 submatrices. In this paper we show that it also holds for the 3 × 3 submatrices, working with the equivalent ‘monomial form’ of the conjecture. This result generalizes to a class of 3 × 3 matrices which will be described.

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on $R^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.

In the second part of the paper we study nonlinear $m$-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for $d\ge 2$, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term approximation can still be derived.

We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are established. As application, we present several new structure relations and non-linear difference equations associated with some of these semi-classical weight functions.

We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite–Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.