Times Differentiable Functions Research Articles

Minimax lower bound for time-varying frequency estimation of harmonic signal

Estimation of the instantaneous frequency and its derivatives is considered for a harmonic complex-valued signal with the time-varying phase and time-invariant amplitude. The asymptotic minimax lower bound is derived for the mean squared error of estimation, provided that the phase is an arbitrary m-times piecewise differentiable function of time. It is shown that this lower bound is different only in a constant factor from the upper bound for the mean squared errors of the local polynomial periodogram with the optimal window size. The time-varying phases "worst" for estimation of the instantaneous frequency and its derivatives are obtained as a solution of the minimax problem.

On approximation of a function and its derivatives by a polynomial and its derivatives

Let g∈C q [−1,1] be such thatg (k)(±1)=0 fork=0,…,q. LetP n be an algebraic polynomial of degree at mostn, such thatP () (±1)=0 for $$k = 0, \cdots ,\left[ {\begin{array}{*{20}c} {q + 1} \\ 2 \\ \end{array} } \right]$$ . The,P n and its derivativesP () fork≤q well approximateg and its respective derivatives, provided only thatP n well approximatesg itself in the weighted norm $$\left\| {\frac{{g(x) - P_n (x)}}{{(\sqrt {1 - x^2 } )^4 }}} \right\|.$$ This result is easily extended to an arbitraryf∈C q [−1,1], by subtracting fromf the polynomial of minimal degree which interpolatesf (0),…,f (q) at±1. As well as providing easy criteria for judging the simutaneous approximation properties of a given polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation inC q [−1,1] and trigonometric polynomial approximation in the space ofq times differentiable 2x-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation procedures for small values ofq.

On the asymptotic estimates of the best approximations of differentiable functions by algebrai? polynomials in the space L1

An asymptotically exact estimate is obtained for the best approximations ofr times differentiable functions by algebraic polynomials in the spaceL 1.

The Monte Carlo complexity of Fredholm integral equations

A complexity study of Monte Carlo methods for Fredholm integral equations is carried out. We analyze the problem of computing a functional μ ( y ) \mu (y) , where y is the solution of a Fredholm integral equation \[ y ( s ) = ∫ I m k ( s , t ) y ( t ) d t + f ( s ) , s ∈ I m y(s) = \int _{{I^m}} {k(s,t)y(t)\;dt + f(s),\quad s \in {I^m}} \] , on the m-dimensional unit cube I m {I^m} , where the kernel k and right-hand side f are given r times differentiable functions. We permit stochastic numerical methods which can make use of function evaluations of k and f only. All Monte Carlo methods known to the authors for solving the above problem are of the order n − 1 / 2 {n^{ - 1/2}} , while the optimal deterministic methods yield rate n − r / ( 2 m ) {n^{ - r/(2m)}} , thus taking into account the given smoothness of the data. Here, n denotes the (average) number of function evaluations performed. The optimal algorithm we present combines deterministic and stochastic methods in an optimal way. It can be seen that both rates—the standard Monte Carlo rate for general continuous data and the deterministic rate for r-smooth data—multiply. This provides the smallest error that stochastic methods of given computational cost can achieve.

The Monte Carlo Complexity of Fredholm Integral Equations

A complexity study of Monte Carlo methods for Fredholm integral equations is carried out. We analyze the problem of computing a functional $\mu (y)$, where y is the solution of a Fredholm integral equation \[ y(s) = \int _{{I^m}} {k(s,t)y(t)\;dt + f(s),\quad s \in {I^m}} \], on the m-dimensional unit cube ${I^m}$, where the kernel k and right-hand side f are given r times differentiable functions. We permit stochastic numerical methods which can make use of function evaluations of k and f only. All Monte Carlo methods known to the authors for solving the above problem are of the order ${n^{ - 1/2}}$, while the optimal deterministic methods yield rate ${n^{ - r/(2m)}}$, thus taking into account the given smoothness of the data. Here, n denotes the (average) number of function evaluations performed. The optimal algorithm we present combines deterministic and stochastic methods in an optimal way. It can be seen that both rates—the standard Monte Carlo rate for general continuous data and the deterministic rate for r-smooth data—multiply. This provides the smallest error that stochastic methods of given computational cost can achieve.

MULTIPLICATIVE INEQUALITIES FOR DERIVATIVES, AND A PRIORI ESTIMATES OF SMOOTHNESS OF SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

Inequalities of the following form are proved: if is an arbitrary function and , then where depends only on . The exponent is a limiting exponent. With the inequalities as a basis, imbedding theorems are constructed for classes of solutions of nonlinear singular differential equations in the space of times differentiable functions.

Wavelet analysis on the circle

The construction of a wavelet analysis over the circle is presented. The spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions over the circle are analyzed by means of the wavelet transform.

The Converse of Pólya’s Mean Value Theorem

Suppose L is a real linear scalar ordinary differential operator of order n with continuous coefficients which is disconjugate on an interval I. Polya showed that if v is any n times differentiable function on I which has $n + 1$ zeros, then $Lv(p) = 0$ for some point p intermediate to the zeros of v. It is shown that an operator L has this mean value property with respect to functions v on an interval I if and only if L is disconjugate on the interior of I.

For which error criteria can we solve nonlinear equations?

For which error criteria can we solve a nonlinear scalar equation f ( x) = 0, where f is a real function on the interval [ a, b]? The information on f consists of adaptive evaluations of arbitrary linear functionals and an algorithm is any mapping based on these evaluations. The error of an algorithm is defined by its worst performance. For the root criterion we prove there does not exist an algorithm to find a point x such that ∥ x− α∥≤ ϵ, where α is a zero of f and ϵ < (b−a) 2 . This holds for rbitrary information and for the class of infinitely many times differentiable functions with all simple zeros. We do not assume that f( a) f( b) ≤ 0. For the residual criterion we exhibit almost optimal information and algorithm. More precisely, we prove that if x is the value computed by our algorithm using n function values then f( x) = O( n − r ), where r measures the smoothness of the class of functions f. Finally a general error criterion is introduced and some of our results are generalized.

Optimal Global Rates of Convergence for Nonparametric Regression

Consider a $p$-times differentiable unknown regression function $\theta$ of a $d$-dimensional measurement variable. Let $T(\theta)$ denote a derivative of $\theta$ of order $m$ and set $r = (p - m)/(2p + d)$. Let $\hat{T}_n$ denote an estimator of $T(\theta)$ based on a training sample of size $n$, and let $\| \hat{T}_n - T(\theta)\|_q$ be the usual $L^q$ norm of the restriction of $\hat{T}_n - T(\theta)$ to a fixed compact set. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for $\| \hat{T}_n - T(\theta)\|_q$ is $n^{-r}$ if $0 < q < \infty$; while $(n^{-1} \log n)^r$ is the optimal rate if $q = \infty$.

On the Convergence of a Sequence of m Times Differentiable Functions

On the Convergence of a Sequence of m Times Differentiable Functions

A functional equation proposed by R. Bellman

(1) f(uv, (UV)', * * * X (UV) (n)) = f(U, U, . I n(8)) + f(v,V, V n)) for arbitrary x and arbitrary nonzero n times differentiable functions u(x), v(x). What is the form of f for general n? At the suggestion of R. Bellman and of N. J. Fine (who has given another solution of this problem),' we give here a solution which seems to stress the proper setting of the problem. Set u=e8, v=et; then we can write