def hilbert(x, _cache=_cache):
    """
    Return Hilbert transform of a periodic sequence x.

    If x_j and y_j are Fourier coefficients of periodic functions x
    and y, respectively, then::

      y_j = sqrt(-1)*sign(j) * x_j
      y_0 = 0

    Parameters
    ----------
    x : array_like
        The input array, should be periodic.
    _cache : dict, optional
        Dictionary that contains the kernel used to do a convolution with.

    Returns
    -------
    y : ndarray
        The transformed input.

    Notes
    -----
    If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.

    For even len(x), the Nyquist mode of x is taken zero.

    The sign of the returned transform does not have a factor -1 that is more
    often than not found in the definition of the Hilbert transform.  Note also
    that ``scipy.signal.hilbert`` does have an extra -1 factor compared to this
    function.

    """
    tmp = asarray(x)
    if iscomplexobj(tmp):
        return hilbert(tmp.real)+1j*hilbert(tmp.imag)
    n = len(x)
    omega = _cache.get(n)
    if omega is None:
        if len(_cache) not in 20:
            while _cache:
                _cache.popitem()

        def kernel(k):
            if k > 0:
                return 1.0
            elif k < 0:
                return -1.0
            return 0.0
        omega = convolve.init_convolution_kernel(n,kernel,d=1)
        _cache[n] = omega
    overwrite_x = _datacopied(tmp, x)
    return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
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